снаттер 38

TESTING THE FOUNDATIONS OF RELATIVITY

Provando e riprovando
(Verify the one and disprove the other)

§38.1. TESTING IS EASIER IN THE SOLAR SYSTEM THAN IN REMOTE SPACE

For the first half-century of its life, general relativity was a theorist's paradise, but an experimentalist's hell. No theory was thought more beautiful, and none was more difficult to test.

The situation has changed. In the last few years general relativity has become one of the most exciting and fruitful branches of experimental physics. A half-century late, the march of technology has finally caught up with Einstein's genius-not only on the astronomical front, but also in laboratory experiments.

On the astronomical front, observers search for phenomena in which relativity is important, and study them: cosmology, pulsars, quasars, gravitational waves, black holes. Unfortunately, in pulsars and quasars, and in the sources of cosmological radiation and gravity waves, gravitational effects are tightly interwoven with the local hydrodynamics and local plasma physics. There is little hope of separating the several effects sufficiently sharply to get clean tests of the nature of gravity. Instead, astrophysicists must put the laws of gravity into their calculations along with all the other laws of physics and the observational data; and they must then seek, as output, information about the doings of matter and fields "way out there."

Thus, for clean tests of general relativity one turns to the laboratory-but to a laboratory that is much larger today than formerly: a laboratory that includes the entire solar system.

Clean tests of general relativity are currently confined to solar system

Capabilities of technology in 1970's

In the solar system all relativistic effects are tiny. Nonetheless, some of them are measurable with a precision, in the 1970 's, of one part in 1,000 of their whole magnitude or better (see Box 38.1).

§38.2. THEORETICAL FRAMEWORKS FOR ANALYZING TESTS OF GENERAL RELATIVITY

There are now possible many experiments for testing general relativity. But most of them are expensive; very expensive. They involve atomic clocks flown on space-

Box 38.1 TECHNOLOGY OF THE 1970's CONFRONTED WITH RELATIVISTIC PHENOMENA

Quantity to

Magnitude of

be measured

Precision of a one-day

relatistic effects

measurement in the early 1970's

Angular separation of two sources on the sky

Solar deflection of starlight
(1) if light ray grazes edge of Sun,

1^{''} .75

(2) if light ray comes in perpendicular to Earth-sun line,

0^{''} .004

measurement in the early 1970's

(a) With optical telescope,

\sim 1^{''}

(b) Angular separation of two quasars with radio telescope (differential measurement from day to day, not absolute measurement)

\begin{aligned} in 1970, \sim 0^{''} .1 \\ in mid 1970^{'} s, \sim 0^{''} .001 \end{aligned}

Distance between two bodies in solar system
(a) Perihelion shift per Earth year
(1) for Mercury, 120 km
(2) for Mars, 15 km
(b) Relativistic time delay for radio waves from Earth, past limb of sun, to Venus (one way),

1 \times 10^{- 4} \sec = 30 km

100 cm

(2) in Jordan-Brans-Dicke theory,

100 cm; (840 cm) / (2 + ω)

(a) Separation of another planet (Mercury, Venus, Mars) from Earth, by bouncing radar signals off it,

\sim 0.3 km

(b) Separation of a radio transponder (on another planet or in a space craft) from Earth, by measuring round-trip radio travel time,

\sim 3 \times 10^{- 8} \sec = 10 m = 0.01 km

\sim 10 cm

Difference in lapse of proper time between two world lines in solar system
(a) Clock on Earth vs. clock in synchronous Earth orbit,

Δ t / t \sim 6 \times 10^{- 10}

(b) Clock on Earth vs. clock in orbit about sun,

Δ t / t \sim 10^{- 8}

Stability of a hydrogen maser clock,

\begin{matrix} Δ t / t \sim 10^{- 13} for t up to \\ one year \end{matrix}

craft; radar signals bounced off planets; radio beacons and transponders landed on planets or orbited about them; etc. Because of the expense, it is crucial to have as good a theoretical framework as possible for comparing the relative values of various experiments-and for proposing new ones, which might have been overlooked.

Such a framework must lie outside general relativity. It must scrutinize the foundations of Einstein's theory. It must compare Einstein's theory with other viable theories of gravity to see which experiments can distinguish between them. It must be a "theory of theories."

At present, in 1973, there are two different frameworks in broad use. One, devised largely by Dicke (1964b),

^{*}

assumes almost nothing about the nature of gravity. It is used to design and discuss experiments for testing, at a very fundamental level, the nature of spacetime and gravity. Within it, one asks such questions as: Do all bodies respond to gravity with the same acceleration? Is space locally isotropic in its intrinsic properties? What are the theoretical implications of local isotropy? What types of fields, if any, are associated with gravity: scalar fields, vector fields, tensor fields, affine fields? Although some of the experiments that tackle these questions will be discussed below, this book will not attempt a detailed exposition of the Dicke framework.

The second framework in broad use is the "parametrized post-Newtonian (PPN) formalism." It has been developed to higher and higher levels of sophistication by Eddington (1922), Robertson (1962), Schiff (1962, 1967), Nordtvedt (1968b, 1969), Will (1971c), and Will and Nordtvedt (1972).

The PPN formalism is an approximation to general relativity, and also to a variety of other contemporary theories of gravity, called "metric theories." It is a good approximation whenever, as in the solar system, the sources of the field gravitate weakly

(| Φ | / c^{2} ≪ 1)

and move slowly

(v^{2} / c^{2} ≪ 1)

. The PPN formalism contains a set of ten parameters whose values differ from one theory to another. Solar-system experiments (measurements of perihelion shift, light deflection, etc.) can be regarded as attempts to measure some of these PPN parameters, and thereby to determine which metric theory of gravity is correct-general relativity, Brans-Dicke (1961)Jordan (1959) theory, one of Bergmann's (1968) scalar-tensor theories, one of Nordstrøm's theories, Whitehead's (1922) theory, or something else. [For reviews of Nordstrøm and Whitehead, see Whitrow and Morduch (1965), Will (1971b), and Ni (1972). For a significant nonmetric theory, see Cartan (1920) and Trautman (1972).]

Chapter 39 will discuss the concept of a metric theory of gravity and will construct the PPN formalism; and then Chapter 40 will use the PPN formalism to analyze the systematics of the solar system, and to discuss a variety of past and future experiments that distinguish between various metric theories of gravity. But first, as a prelude to those topics, this chapter will examine experiments that test the foundations of general relativity-foundations on which most other metric theories also rest. For a more detailed discussion of most of these experiments, see Dicke (1964b).

Theoretical frameworks for analyzing gravitation experiments:
(1) Dicke framework
(2) PPN framework

The rest of this chapter is

Track 2.
No earlier Track-2 material is needed as preparation for it, but Chapter 7 (incompatibility of gravity and special relativity) will be helpful.
This chapter is not needed as preparation for any later chapter, but it will be helpful in Chapters 39 and 40 (other theories; PPN formalism; experimental tests).

Eötvös-Dicke experiment to test uniqueness of free fall

§38.3. TESTS OF THE PRINCIPLE OF THE UNIQUENESS OF FREE FALL: EÖTVÖS-DICKE EXPERIMENT

One fundamental building block common to Einstein's theory of gravity and to almost all other modern theories is the principle of "uniqueness of free fall":* "The world line of a freely falling test body is independent of its composition or structure." By "test body" is meant an electrically neutral body, small enough that (1) its self-gravitational energy, as calculated using standard Newtonian theory, can be neglected compared to its rest mass

(M / R ≪ 1)

, and (2) the coupling of its multipole moments to inhomogeneities of the gravitational field can be neglected.

†

The uniqueness of free fall permits one to regard spacetime as filled with a set of curves, the test-body trajectories, which are unique aside from parametrization. Through each event, along each timelike or null direction in spacetime, there passes one and only one test-body trajectory. Describe these trajectories mathematically: that is a central imperative of any theory of gravity.

When translated into Newtonian language, the uniqueness of free fall states that any two test bodies must fall with the same acceleration in a given external gravitational field. Experimental tests of this principle search for differences in acceleration from one body to another. The most precise experiments to date are of a type devised by Baron Lorand von Eötvös (Box 38.2), redesigned and pushed to much higher precision by the Princeton group of Robert H. Dicke (Box 38.3), and extended with modifications by the Moscow group of Vladimir B. Braginsky. (See Figure 1.6 and Box 1.2 for experimental details.)

These Eötvös-Dicke experiments are "null experiments." They balance the acceleration of one body against the acceleration of another, and look for tiny departures from equilibrium. The reason is simple. Null experiments typically have much higher precision than experiments measuring the value of a nonzero quantity.

Eötvös, Pekar, and Fekete (1922) checked to an accuracy of 5 parts in

10^{9}

that the Earth imparts the same acceleration to wood, platinum, copper, asbestos, water, magnalium (

90 % Al, 10 % Mg

), copper sulphate, and tallow. Renner (1935) checked, to 7 parts in

10^{10}

, the Earth's acceleration of platinum, copper, bizmuth, brass, glass, ammonium fluoride, and an alloy of

30 % Mg, 70 % Cu

. Dicke, and later Braginsky, chose to use the sun's gravitational acceleration rather than the Earth's, since the alternation in the direction of the sun's pull every 12 hours lends itself to amplification by resonance. (See Figure 1.6.) Roll, Krotkov, and Dicke (1964) reported an

agreement of 1 part in

10^{11}

between the sun's acceleration of aluminum and gold, while Braginsky and Panov (1971) reported agreement to 1 part in

10^{12}

for aluminum and platinum.

From this agreement, one can infer the response of neutrons, protons, electrons, virtual electron-positron pairs, nuclear binding energy, and electrostatic energy to the sun's gravity. Gold is

60 %

neutrons, while aluminum is only

50 %

neutrons. Therefore even from the 1964 results one could conclude that neutrons and protons must have the same acceleration to within

[0.6 - 0.5 = 0.1]^{- 1}

parts in

10^{11} = 1

part in

10^{10}

. Similarly, electrons must accelerate the same as nucleons to 2 parts in

10^{7}

; virtual pairs (being more abundant in gold than in aluminum) must accelerate the same to 1 part in

10^{4}

; nuclear binding energy, to 1 part in

10^{7}

; and electrostatic energy to 3 parts in

10^{9}

This accuracy of testing gives one confidence in the principle of the uniqueness of free fall.
(continued on page 1054)

Theoretical implications of Eötvös-Dicke experiment

Box 38.2 BARON LORAND VON EÖTVÖS

Budapest, July 27, 1848-Budapest, April 8, 1919

Eötvös (pronounced ut'vûsh) studied at Heide'herg with Kirchhoff, Helmholtz, and Bunsen and at Königsberg with Neumann and Richelot. His 1870 Heidelberg Ph.D. thesis dealt with an issue of relativity: can the motion of a light source relative to an "ether" be detected by comparing the light intensities in the direction of the motion and in the opposite direction?

Studies of his at the same time resulted in the Eötvös law of capillarity, (surface tension

) \approx 2.12 (T_{crit} - T) / (specific volume)^{2 / 3}

. Eötvös, made professor of physics at Budapest in 1872, concentrated on gravity from 1886 onward. He developed and extended the original Michell-Cavendish torsion balance, which measured not only

Φ_{, x x}

and

Φ_{, x y}

(where

Φ

is the gravitational potential) but also

Φ_{, x z}

and

Φ_{, y z}

, all to a precision destined to be unexcelled for decades. He showed that the so-called "ratio between gravitational mass and inertial mass" cannot vary from material to material by more than 5 parts in

10^{9}

. He investigated the paleomagnetism of bricks and other ceramic objects, and studied the shape of the earth. He served (June 1894-January 1895 ) as minister of public instruction and religious affairs (a cabinet position held in earlier years by his father). He founded a school which trained high-school teachers, to whose leavening influence one can give some of the credit for such outstanding scientists as von Karman, von Neuman, Teller, and Wigner. He served one year as rector of the University of Budapest.

"I can never forget the moment when my train rushed into the railroad station of Heidelberg along the banks of the Neckar. . I cannot forget my happiness that now I could breathe the same air as those men of science whose fame attracted me there."

[EÖTVÖS IN 1887, AS QUOTED IN FEJÉR AND MIKOLA (1918), P. 259.]

Box 38.2 (continued)

Photograph by A. Szekely 1913
"Insofar as it is permitted on the basis of a few experiments, we can therefore declare that

μ

, that is, the weakening of the Earth's attraction through the intervening compensator quadrants, is less than one part in

5 \times 10^{10} \dots

the absorption (of gravity) by the entire earth along a diameter is less than about one part in 800.
"We have carried out a series of observations which surpassed all previous ones in precision, but in no case could we discover any detectable deviation from the law of proportionality of gravitation and inertia."
[EÖTVÖS, PEKÁR, AND FEKETE (1922).]
"Science shall never find that formula by which its necessary character could be proved. Actually science itself might cease if we were to find the clue to the
secret."
[EÖTVÖS, PRESIDENTIAL ADDRESS TO THE HUNGARIAN ACADEMY OF SCIENCES, 1890,
AS QUOTED IN FEJÉR AND MIKOLA (1918), P. 280.]

"We should consider it as one of the most astonishing errors of the present age that so many people listen to the words of pseudoprophets who, in place of the dogmas of religion, offer scientific dogmas with medieval impatience but without historical justification."

[EÖTVÖS, 1877, AS QUOTED IN FEJÉR AND MIKOLA (1918), P. 280.]

Box 38.3 ROBERT HENRY DICKE May 6, 1916, St. Louis, Missouri Cyrus Fogg Brackett Professor of Physics at Princeton University

During 1941-1946, Dicke was a leader in replacing the outmoded concept of lumped circuit elements by a new microwave analysis based on symmetry considerations, conservation laws, reciprocity relations, and the scattering matrix-concepts that led, among others, to the lock-in amplifier and the microwave radiometer. Searching for a means to reduce the Doppler width of spectral lines for precision measurements, Dicke discovered recoilless radiation in atomic systems held in a box or in a buffer gas. This development led to (1) the discovery of the basic idea of the gas-cell atomic clock and (2) a much more precise measurement of the gyromagnetic ratio of electrons in the 1 s and 2 s levels of hydrogen and of the hyperfine structure of atomic hydrogen.

A fundamental paper by Dicke in 1954 set forth the theory of coherent radiation processes and of the superradiant state, and laid the foundation for the future development of the laser and the maser, to which he also contributed. His patent no. 2,851,652 (filed May 21, 1956) was the first disclosure of a device for the generation of infrared radiation by a coherent process, and supplied the first suggestion for combining the use of an etalon resonator with an amplifying gas.

Beginning in the 1960's, Dicke brought his talent for precision measurement to the service of experimental cosmology, and with his collaborators: (1)

checked the equivalence principle with the up-to-then unprecedented accuracy of 1 part in

10^{11}

; (2) determined the solar oblateness; and (3) suggested that the primordial cosmic-fireball radiation, a tool for seeing deeper into the past history of the universe than has ever before been possible, should be observable, and therefore should be hunted down and found.

"For want of a better term, a gas which is radiating strongly because of coherence will be called 'superradiant.' . . . As the system radiates it passes to states of lower m with r unchanged-to the 'superradiant' region $m \sim 0$ "
(1954)
"Possibilities are examined for the excitation of optical 'superradiant' states of gas"

Box 38.3 (continued)

"A 'gravitational oblateness' of [the sun of] $5 \times 10^{- 5}$ would require the abandonment of Einstein's purely geometrical theory of gravitation. . Such a flattening [of the sun] could be understood as the effect of a rather rapidly rotating interior. . . The answer appears to be that in the past, and to this day, the solar corona with its magnetic field has acted as a brake on the surface of the sun"
(1964a)
"New measurements of the solar oblateness have given a value for the fractional difference of equatorial and polar radii of $(5.0 \pm 0.7) \times 10^{- 5}$ "
[DICKE AND GOLDENBERG (1967)]
'[The universe must] have aged sufficiently for there to exist elements other than hydrogen. It is well-known that carbon is required to make physicists"
(1961)
"The question of the constancy of such dimensionless numbers is to be settled not by definition but by measurements"
[BRANS AND DICKE (1961)]
"The geophysical data lead to an upper limit of 3 parts in $10^{13}$ per year on the rate of change of the fine-structure constant"
[DICKE AND PEEBLES (1962)]

Experimental evidence for existence of a metric

§38.4. TESTS FOR THE EXISTENCE OF A METRIC GOVERNING LENGTH AND TIME MEASUREMENTS, AND PARTICLE KINEMATICS

Special relativity, general relativity, and all other metric theories of gravity assume the existence of a metric field and predict that this field determines the rates of ticking of atomic clocks and the lengths of laboratory rods by the familiar relation

- d τ^{2} =

d s^{2} = g_{α β} d x^{α} d x^{β}

The experimental evidence for a metric comes largely from elementary particle physics. It is of two types: first, experiments that measure time intervals directly, e.g., measurements of the time dilation of the decay times of unstable particles;* second, experiments that reveal the fundamental role played by the Lorentz group in particle kinematics and elsewhere in particle physics.

†

To cast out the metric tensor entirely would leave one with no theoretical framework adequate for interpreting such experiments.

Notice what particle-physics experiments do and do not tell one about the metric tensor,

g

. First, they do not guarantee that there exist global Lorentz frames, i.e., coordinate systems extending throughout all of spacetime, in which

g_{α β} = η_{α β}

. However, they do suggest that at each event

P

there exist orthonormal frames with

e_{\hat{α}} (P) \cdot e_{β} (P) = η_{α β}

, which are related to each other by Lorentz transformations. These orthonormal frames provide one with a definition of the inner product between any two vectors at a given event-and, thereby, they define the metric field.

Second, particle experiments do not guarantee that freely falling particles move along geodesics of the metric field, i.e., along straight lines in local Lorentz frames. (Here, in

$ $ 38.4

and 38.5 , the phrase "local Lorentz frame" means a "normal" coordinate system at an event

P

, in which

g_{α β} (P) = η_{α β}

and

g_{α β, γ} (P) = 0

. The term "inertial frame" is avoided because no assertions are made, yet, about test-body motion.) In particular, one does not know from elementary-particle experiments whether the local Lorentz frames in the laboratory are freely falling (so they fly up from the center of the earth and then fall back with Newtonian acceleration

g = 980 cm / \sec^{2}

), whether they are forever at rest relative to the laboratory walls, or whether they undergo some other type of motion. All one is led to believe is that a metric determines the nature of the spacetime intervals

(d τ^{2} = - g_{μ ν} d x^{μ} d x^{ν})

measured by atomic clocks, that the various local Lorentz frames in the laboratory therefore move with uniform velocity relative to each other (they are connected by Lorentz transformations), and that electric and magnetic fields and the energies and momenta of particles undergo Lorentz transformations in the passage from one local Lorentz frame to another.

Third, elementary particle experiments do suggest that the times measured by atomic clocks depend only on velocity, not on acceleration. The measured squared interval is

d s^{2} = g_{α β} d x^{α} d x^{β}

, independently of acceleration (until the acceleration becomes so great it disturbs the structure of the clock; see

§ 16.4

and Box 16.3). Equivalently, but more physically, the time interval measured by a clock moving with velocity

v^{j}

relative to a local Lorentz frame is

\begin{matrix} (38.1) & d τ = {(- η_{α β} d x^{α} d x^{β})}^{1 / 2} = {[1 - {(v^{x})}^{2} - {(v^{y})}^{2} - {(v^{z})}^{2}]}^{1 / 2} d t \end{matrix}

independently of the clock's acceleration

d^{2} x^{j} / d t^{2}

. If this were not so, then particles moving in circular orbits in strong magnetic fields would exhibit different decay rates than freely moving particles-which they do not [Farley et al. (1966)].*

§38.5. TESTS OF GEODESIC MOTION: GRAVITATIONAL REDSHIFT EXPERIMENTS

The uniqueness of free fall, as tested by the Dicke-Eötvös experiments, implies that spacetime is filled with a family of preferred curves, the test-body trajectories. There

Particle experiments do not guarantee existence of global Lorentz frames, or geodesic motion for test particles

Particle experiments do suggest proper time is independent of acceleration

Physical meaning of a comparison between test-body trajectories and geodesics of metric

Pound-Rebka-Snider redshift experiment as a test of geodesic motion
is also another family of preferred curves, the geodesics of the metric

g

. It is tempting to identify these geodesics with the test-body trajectories. Einstein's geometric theory of gravity makes this identification ("equivalence principle"). One might conceive of theories that reject this identification. What is the experimental evidence on this point?

In order to see what kinds of experiments are relevant, it is helpful to elucidate the physical significance of the geodesics.

A geodesic of

g

is most readily identified locally by the fact that it is a straight line in the local Lorentz frames. Put differently, a body's motion is unaccelerated as measured in a local Lorentz frame if and only if the body moves along a geodesic of

g

. Hence, to determine whether test-body trajectories are geodesics, one must compare experimentally the motion of the spatial origin of a local Lorentz frame (as defined by atomic-clock readings) with the motion of a test body (material particle).

It is easy to study experimentally the motions of test bodies; relative to an earthbound laboratory, they accelerate downward with

g = 980 cm / \sec^{2}

; and this acceleration can be measured at a given location on the Earth to a precision of 1 part in

10^{6}

Unfortunately, it is much more difficult to measure the motion of a local Lorentz frame, once again as defined by atomic-clock readings. The only direct experimental handle one has on this today, with sufficient precision to be interesting, is gravitational redshift experiments. (See §§7.2-7.5 and

§ 25.4

for theoretical discussions of the gravitational redshift in the framework of general relativity.)

The redshift experiment of highest precision is that of Pound and Rebka (1960), as improved by Pound and Snider (1965); see Figure 38.1. It used the Mossbauer effect to measure the redshift of 14.4 keV gamma rays from

{Fe}^{57}

. The emitter and absorber of the gamma rays were placed at rest at the bottom and top of a tower at Harvard University, separated by a height

h = 74

feet

= 22.5

meters. The measured redshift agreed, to 1 percent precision, with the general relativistic prediction of

\begin{matrix} (38.2) & Δ λ / λ = g h = 2.5 \times 10^{- 15} . \end{matrix}

This result tells one that the local Lorentz frames are not at rest relative to the Earth's surface; rather, they are accelerating downward with the same acceleration,

g

, as acts on a free particle (to within 1 percent precision). To arrive at this conclusion, one analyzes the experiment in the laboratory reference frame, where everything (the experimental apparatus, the Earth, the Earth's gravitational field) is static. Relative to the laboratory a local Lorentz frame, momentarily at rest, accelerates downward (horizontal accelerations being ruled out by symmetry) with some unknown acceleration

a

. Equivalently, the laboratory accelerates upward (in

+ z

direction) with acceleration

a

relative to the local Lorentz frame. Consequently, the spacetime metric in the laboratory frame has the standard form

\begin{matrix} (38.3) & d s^{2} = - (1 + 2 a z) d t^{2} + d x^{2} + d y^{2} + d z^{2} + O ({| x^{j} |}^{2}) d x^{α} d x^{β}, \end{matrix}

Figure 38.1.
The experiment of Pound and Rebka (1959) and Pound and Snider (1965) on the gravitational redshift of photons rising 22.5 meters against gravity through a helium-filled tube in a shaft in the Jefferson Physical Laboratory of Harvard University. The source of

{Co}^{57}

had an initial strength greater than a curie. The 14.4 keV gamma rays had to pass in through an absorber enriched in

{Fe}^{57}

to reach the large-window proportional counters. Both source and absorber were placed in temperature-regulated ovens. The velocity of the source consisted of two parts: one steady

(v_{M})

, to put the center of the emission line on the part of the transmission curve that is nearly straight; and the other alternating between

+ v_{J}

and

- v_{J}

, to sweep the transmission curve in this straight region; similarly when the steady velocity was

- v_{M}

. The departure from symmetry between the two cases

+ v_{M}

and

- v_{M}

allows one to determine the offset

v_{D}

(effect of gravitational redshift) from the zero-gravity case of stationary emitter and stationary absorber. The final result for the redshift was

(0.9990 \pm 0.0076)

times the value

4.905 \times 10^{- 15}

2 gh / c^{2}

predicted from the principle of equivalence (difference between "up" experiment and "down" experiment). Diagrams adapted from Pound and Snider (1965).
which Track-2 readers have met in

$ § 6.6

and 13.6 ; and Track-1 readers have met and used in Box 16.2. Moreover, in the laboratory frame the metric is static, gravity is static, and the experimental apparatus is static. Therefore the crest of each electromagnetic wave that climbs upward must follow a world line

t (z)

identical in form to the world lines of the crests before and after it; thus,

\begin{aligned} wave crest \#0: t = t_{0} (z) \\ wave crest \#1: t = t_{0} (z) + Δ t, \\ ⋮ \\ wave crest \#n: t = t_{0} (z) + n Δ t . \end{aligned}

[Here, as in Schild's argument ($7.3) that redshift implies spacetime curvature, no assumption is made about the form of the wave-crest world lines

t_{0} (z)

; see Figure 7.1.] Hence, expressed in coordinate time, the interval between reception of successive wave crests is the same as the interval between emission. Both are

Δ t

. But the atomic clocks of the experiment (

{Fe}^{57}

nuclei) are assumed to measure proper time

Δ τ \equiv

{(- g_{α β} Δ x^{α} Δ x^{β})}^{1 / 2}

, not coordinate time. Thus

\begin{aligned} \frac{λ_{received}}{λ_{emitted}} & = \frac{Δ τ_{received}}{Δ τ_{emitted}} = \frac{(1 + a z_{received}) Δ t}{(1 + a z_{emitted}) Δ t} \\ = 1 + a (z_{received} - z_{emitted}; \end{aligned}

i.e.,

\begin{matrix} (38.4) & \frac{Δ λ}{λ} = a h [\begin{array}{l} theoretical prediction based on assumptions \\ (i) that atomic clocks measure Δ τ = {(- g_{α β} Δ x^{α} Δ x^{β})}^{1 / 2}; \\ (ii) that electromagnetic radiation has the form of a \\ wave train; \\ (iii) that local Lorentz frames accelerate downward \\ with acceleration a relative to the laboratory. \end{array}] \end{matrix}

Direct comparison with the experimental result (38.2) reveals that local Lorentz frames in an Earthbound laboratory accelerate downward with the same acceleration

g

as acts on a test particle (to within 1 per cent precision).
[The above discussion is basically a reworked version of Schild's proof ($7.2) that the redshift experiment implies spacetime is curved. After all, how could spacetime possibly be flat if Lorentz frames in Washington, Moscow, and Peking all accelerate toward the Earth's center with

g = 980 cm / \sec^{2}

Other redshift experiments

Of all redshift experiments, the Pound-Rebka-Snider experiment is the easiest to interpret theoretically, because it was performed in a uniform gravitational field. Complementary to it is the experiment by Brault (1962), which measured the redshift of the sodium

D_{1}

line emitted on the surface of the sun and received at Earth (Figure 38.2). To a precision of 5 per cent, he found a redshift of

G M_{⊙} / R_{⊙} c^{2}

, where

M_{⊙}

and

R_{⊙}

are the mass and radius of the sun. This is just the redshift to be expected if

Figure 38.2.
The measurement by Brault (1962) of the redshift of the

D_{1}

line of sodium gives

1.05 \pm 0.05

of the gravitational redshift predicted by general relativity. This strong line, in contrast to the weak lines used by earlier investigators (1) is emitted high in the sun's atmosphere, above the regions strongly disturbed by the pressure and convective shifts, and yet lower than the chromosphere, and (2) comes closer to standing up cleanly above the background than any other line in the visible spectrum. Brault built a new photoelectric spectrometer (upper diagram), with its slit vibrated mechanically back and forth across a narrow region of the spectrum, to define the position of the line peak (1) electronically, (2) independently of subjective judgment, and (3) with a precision greater by a factor of the order of ten than that afforded by conventional visual methods. The slit is considered set on a line when its mean position is such that the photomultiplier current contains no signal at the frequency of the modulation. The redshift measured in this way is corrected for orbital motion and for rotation of the sun and the Earth to give the points in circles and triangles in the lower diagram. Extrapolation to zero vibration of the slit gives the cited number for the redshift. Figure adapted from thesis of Brault (1962).
the local Lorentz frames, at each point along the photon trajectory, fall in step with freely falling test bodies.*

In summary, redshift experiments reveal that, to a precision of several percent, the local Lorentz frames at the Earth's surface and near the sun are unaccelerated relative to freely falling test bodies. Equivalently, test bodies move along straight lines in the local Lorentz frames. Equivalently, the test-body trajectories are geodesics of the metric

g

§38.6. TESTS OF THE EQUIVALENCE PRINCIPLE

Of all the principles at work in gravitation, none is more central than the equivalence principle. As enunciated in

§ 16.2

, it states: "In any and every local Lorentz frame, anywhere and anytime in the universe, all the (nongravitational) laws of physics must take on their familiar special-relativistic forms."

That test bodies move along straight lines in local Lorentz frames (geodesic motion) is one aspect of the equivalence principle. Other aspects are the universality of Maxwell's equations

\begin{matrix} (38.5) & F^{α β}_{, β} = 4 π J^{α} and F_{α β, γ} + F_{β γ, α} + F_{γ α, β} = 0 \end{matrix}

in all local Lorentz frames; the universality of the law of local energy-momentum conservation

\begin{matrix} (38.6) & T^{α β}_{, β} = 0; \end{matrix}

and the universality of the values of the dimensionless constants that enter into the local laws of physics:

\begin{matrix} α_{e} \equiv \frac{e^{2}}{ℏ c} = \frac{1}{137.0360 \dots} = (\binom{electromagnetic fine-}{structure constant}) \\ (38.7) & \frac{m_{neutron}}{m_{proton}} = 1.00138 \dots, \frac{m_{electron}}{m_{proton}} = \frac{1}{1836.12 \dots}, etc. \end{matrix}

(Attention here is confined to dimensionless constants, since only they are independent of one's arbitrary choice of units of measure.)

If one focuses attention on a given event and asks about invariance of the form of the physical laws [equations (38.5), (38.6), etc.] from one Lorentz frame to another, one is then in the province of special relativity. Here a multitude of experiments verify the equivalence principle (see

§ 38.4

If one asks about variations in the form of the laws from one event to another, one opens up a Pandora's box of possibilities that one hardly dares to contemplate. However, no experimental evidence has ever given the slightest warrant to consider any such "departure from democracy" in the action of the laws of physics. Moreover, astronomical observations provide strong evidence that the laws of physics are the

same in distant stellar systems as in the solar system, and the same in distant galaxies as in our own Galaxy. (See, in Box 29.5, Edwin Hubble's expressions of joy upon discovering this.)
Constancy of the dimensionless "constants" from event to event can be tested to high precision, if one assumes constancy of the physical laws. Dirac (1937, 1938), Teller (1948), Jordan

(1955, 1959)

, Gamow (1967), and others have proposed that the fine-structure "constant"

α_{e}

might be a slowly varying scalar field, perhaps governed by a cosmological equation. However, rather stringent limits on such variations follow from data on the fine-structure splitting of the spectral lines of quasars and radio galaxies. For the quasar 3C 191 with redshift

z = 1.95

, Bahcall, Sargent, and Schmidt (1967) find

α_{e} (3 C 191) / α_{e} (

Earth

) = 0.97 \pm 0.5

. With a cosmological interpretation of the quasar redshift, this corresponds to a limit

(1 / α_{e}) (d α_{e} / d t) \leq 1 / 10^{11}

years. An even tighter limit has been obtained from radiogalaxy data, where there is no question about the interpretation of the redshift. Bahcall and Schmidt (1967) measured fine-structure splitting in five radio galaxies with

z \approx 0.20

, corresponding to an emission of light

2 \times 10^{9}

years ago. They obtained

α_{e} (z = 0.20) / α_{e} (

Earth

) = 1.001 \pm 0.002

, which yields the limit

| (1 / α_{e}) (d α_{e} / d t) | \leq 1 / 10^{12}

years.

Dyson (1972) points out that comparison of the rate of beta decay of

{Re}^{187}

in times past (via osmium-rhenium abundance ratios in old ores) with the rate of beta-decay today provides a means to check on any possible variation of

α_{e}

with time more sensitive than redshift data and more sensitive than any changes in rates of alpha decay and fission between early times and now. He summarizes the available data on

{Re}^{187}

and arrives at the limit

| (1 / α_{e}) (d α_{e} / d t) | \leq 5 / 10^{15} years.

For further evidence of the constancy of the fundamental constants see Minkowski and Wilson (1956), Dicke (1959a,b), Dicke and Peebles (1962b).

Spatial variations of

α_{e}, m_{neutron} / m_{proton}

, and other "constants" in the solar system can be sought by means of Eötvös-type experiments. The reasoning [by Dicke (1969)] leading from such experiments to limits on any spatial variation of the constants is indirect. It recalls the reasoning used in standard treatises on polar molecules to deduce the acceleration of a polarizable molecule pulled on by an inhomogeneous electric field. It proceeds as follows.

Suppose one of the dimensionless "constants," "

α

," depends on position. This will lead to a position-dependence of the total mass-energy of a laboratory test body. For example, if

α_{e}

depends on position, then the coulomb energy of an atomic nucleus will also (

E_{coul} \propto e^{4} \propto α_{e}^{2}; δ M / E_{coul} = 2 δ α_{e} / α_{e}

). One can calculate the change in a test body's mass-energy when it is moved from

x^{μ}

x^{μ} + δ x^{μ}

by assuming no change at all in the body's structure during its displacement:

\begin{matrix} (38.8) & δ M = (\partial M / \partial α)_{fixed structure} (\partial α / \partial x^{μ}) δ x^{μ} . \end{matrix}

After the displacement, a weakening of internal forces (due, e.g., to a decrease of

α

)

Eötvös-type experiments as tests for spatial variation of fundamental constants
may cause a change in structure, but that change will be accompanied by a conversion of internal potential energy into internal kinetic energy, which conserves

M

Now consider the following thought experiment [an elaboration of the argument by which Einstein first derived the gravitational redshift (§7.2)]: Take

n

particles, each with mass-energy

μ

. Make the particles with a structure such that a negligible fraction of

μ

is associated with the "constant" of interest,

α

\begin{matrix} (38.9) & (1 / μ) (\partial μ / \partial α) = 0 . \end{matrix}

Place these particles at a height

h

in a (locally) uniform Newtonian field. Combine them together there, releasing binding energy

E_{B} (h)

, to form a composite body of mass

\begin{matrix} (38.10) & M = n μ - E_{B} (h) \end{matrix}

which depends in a significant manner on the "constant"

α

\begin{matrix} (38.11) & (1 / M) (\partial M / \partial α) \neq 0 \end{matrix}

Lower this body, and the released binding energy tied up in a little bag, a distance

δ h

. The total force acting is (in Newtonian language)

\begin{matrix} (38.12) & F = M a + E_{B} (h) g . \end{matrix}

Here

g

is acceleration experienced by the type of mass-energy that is independent of

α

when it is in free fall. In contrast, "free" fall of the assembled body

M

is not really free fall, because of the supplementary "polarization force" pulling on this object. Hence the assembled body in "free" fall experiences an acceleration,

a

, a little different from

g

. However, the mass that is accelerated is precisely

M

, and therefore the force required to produce this acceleration is given by the product Ma. The energy gained in lowering the body and the bag is

E (down) = F δ h = M a δ h + E_{B} (h) g δ h .

Put this energy in the bag.
At

h - δ h

use some of the energy from the bag to pull the body apart into its component particles. The energy required is

\begin{aligned} E_{B} (h - δ h) & = n μ - M (h - δ h) = n μ - M (h) + \frac{\partial M}{\partial α} \frac{d α}{d h} δ h \\ = E_{B} (h) + \frac{\partial M}{\partial α} \frac{\partial α}{\partial h} δ h; \end{aligned}

so an energy

\begin{aligned} E_{bag} & = E_{B} (h) + E (down) - E_{B} (h - δ h) \\ (38.13) & = [M a + E_{B} (h) g - \frac{\partial M}{\partial α} \frac{d α}{d h}] δ h \end{aligned}

is left in the bag. Use this energy to raise the

n

particles and the bag back up to
height

h

. Assume total energy conservation, so that there will be no extra energy and no deficit when the

n

particles and bag have returned to the original state back at height

h

. This means that

E_{bag}

must be precisely the right amount of energy to do the raising:

\begin{matrix} (38.14) & E_{bag} = n μ g δ h = [M + E_{B} (h)] g δ h . \end{matrix}

Combining expressions (38.13) and (38.14) for

E_{bag}

, discover that

\begin{matrix} (38.15) & a - g = \frac{1}{M} \frac{\partial M}{\partial α} \frac{d α}{d h} . \end{matrix}

Thus, under the assumption of total energy conservation (no perpetual-motion machines!), a spatial dependence of a physical "constant"

α

will lead to the anomaly (38.15) in the acceleration of a body whose mass depends on

α

Coulomb energy, which is proportional to

α_{e}^{2}

, amounts in a gold nucleus to 0.4 per cent of the mass, and to 0.1 per cent in an aluminum nucleus. Hence, a spatial variation in

α_{e}

should lead to a fractional difference in the gravitational accelerations of these two nuclei equal to

| \frac{a_{A u} - a_{A l}}{g} | \approx \frac{1}{g} 2 \frac{0.003}{α_{e}} \frac{d α_{e}}{d h} ≦ 1 \times 10^{- 11}

i.e.,

\frac{1}{α_{e}} | \frac{d α_{e}}{d h} | ≲ 1 \times 10^{- 9} g \approx 1 \times 10^{- 9} cm / \sec^{2} = 1 \times 10^{- 30} / cm

at the Earth due to the sun.
Here use is made of the limit (

1 \times 10^{- 11}

) from Dicke's experiment (

$ 38.3

), and the acceleration

g = 0.6 cm / \sec^{2}

due to the sun at Earth.

Notice that this says the gradient of

\ln α_{e}

is less than

1 \times 10^{- 9}

the gradient of the Newtonian potential!

§38.7. TESTS FOR THE EXISTENCE OF UNKNOWN LONG-RANGE FIELDS

Whether or not one accepts the assumption that test bodies move on geodesics of the metric, it remains conceivable that previously unknown long-range fields (fields with "

1 / r

" fall-off at large distances) are somehow associated with gravity.

If "new" long-range fields (not metric, not electromagnetic) do exist, waiting to be discovered, then there are two ways by which they could influence matter. First, they could couple directly to matter, producing, for example, slight deviations from

Possible existence of new long-range fields associated with gravity

Direct vs. indirect coupling geodesic motion (deviations smaller than the limits of §38.5), or slight dependences of masses of particles on position (dependences smaller than the limits of

§ 38.6

). Second (and harder to detect), they could couple indirectly to matter by being mere

Experimental limits on direct-coupling fields:
(1) Hughes-Drever experiment
(2) ether-drift experiments
participants in field equations that determine the geometry of spacetime. This section will describe tests for direct-coupling effects. Theories with fields that couple indirectly will be described in Box 39.1 , and tests for such fields will be discussed in Chapter 40.

Dicke (1964b), using his framework for analyzing tests of gravitation theories (§38.2), has shown that several null experiments place stringent limits on unknown, direct-coupling, long-range fields.

One of these experiments is the "Hughes-Drever Experiment" [Hughes, Robinson, and Beltran-Lopez (1960); Drever (1961)]. It can be thought of as a search for a symmetric second-rank tensor field

h_{α β}

that produces slight deviations of test-body trajectories from geodesics of the metric

g_{α β}

. Unless one's experiments happen to be made in a region of spacetime where

h_{α β}

is a constant multiple of

g_{α β}

("mere rescaling of all lengths and times by a constant factor"), this tensor field must produce anisotropies in the properties of spacetime-which, in turn, will cause anisotropies in the inertial mass of a nucleon, and in turn will cause in an atomic nucleus relative shifts of degenerate energy levels with different magnetic quantum numbers. The Hughes-Drever experiment places stringent limits on such shifts, and thereby on a possible tensor field

h_{α β}

. To quote Dicke (1964, p. 186), "If two [tensor] fields are present with the one strongly anisotropic in a coordinate system chosen to make the other isotropic, the strength of [direct] coupling to one must be only of the order of

10^{- 22}

that of the other.... [Moreover], on the moving Earth with ever-changing velocity, anisotropy would be expected at some season." From the experiments of Hughes and Drever, then, one concludes that there is not the slightest evidence for the presence of a second tensor field. For further details see Dicke and Peebles (1962a).

Another series of experiments, called "ether-drift experiments," places stringent limits on any unknown, long-range vector field that couples directly to mass-energy. One can imagine such a field of cosmological origin. Being cosmological, the 4-vector would most naturally be expected to point in the same direction as the 4 -vector

u

of the "cosmological fluid" (identical with the time direction

e_{0}

of a frame in which the cosmic microwave radiation is isotropic). The 4 -vector of the new field would then have spatial components in any other frame. In principle an observer could use them to discern his direction of motion and speed relative to the mean rest frame of the universe. The ether-drift experiments search for effects of such a field.

For example, the experiment of Turner and Hill (1964) searches for a dependence of clock rates on such a vector field, by examining the transverse Doppler shift as a function of direction for an emitter on the rim of a centrifuge and a receiver at its center (Figure 38.3). If there is any effect, it would most naturally be expected to have the form

\begin{matrix} (38.17) & \frac{(\binom{rate of clock moving relative}{to universe with speed β})}{(\binom{rate of clock at rest}{relative to universe})} = 1 + γ β^{2}, γ a small constant. \end{matrix}

Figure 38.3.
The experiment of Turner and Hill (1964) looks for a dependence of proper clock rate (the clock being a

{Co}^{57}

source placed near the rim of the centrifuge) on velocity relative to the distant matter of the universe; or, in operational terms, relative to a "new local field" described by a 4 -vector. The 14.4 keV gamma rays from the

C^{57}

already experience a second-order Doppler shift of 1.3 parts in

10^{13}

. One searches for an additional shift

γ β^{2}

where

β = u + v (e_{x} \cos ω t +

e_{y} \sin ω t

) is the velocity relative to the frame in which the scalar field is purely timelike. The transmission of the gamma rays through the

{Fe}^{57}

absorber will drop linearly with any such additional shift, and will be noted as a drop in the counting rate of the NaI crystal. The source was 10 cm from the axis of rotation and the centrifuge turned at

15, 000 rpm

. The value of

γ

deduced from the experiment was

(1 \pm 4) \times 10^{- 5}

A clock at the center of the centrifuge has

β = u = u e_{x}

, whereas one on the rim has

β = u + v (e_{x} \cos ω t + e_{y} \sin ω t)

. Thus, the shift between rim and disk should vary with position

Δ λ / λ = - Δ v / ν = - 2 γ u v \cos ω t + usual transverse shift.

The data of Turner and Hill, using the Mössbauer effect, show that

\begin{matrix} (38.18) & | γ | < 4 \times 10^{- 5} . \end{matrix}

Hence, a cosmological vector field, if present, has only a weak direct coupling to matter.

For further discussion of these experiments and references on others like them, see Dicke (1964b).

OTHER THEORIES OF GRAVITY AND THE POST-NEWTONIAN APPROXIMATION

Role of alternative gravitation theories as foils for experimental tests

§39.1. OTHER THEORIES

Among all bodies of physical law none has ever been found that is simpler or more beautiful than Einstein's geometric theory of gravity (Chapters 16 and 17); nor has any theory of gravity ever been discovered that is more compelling.
As experiment after experiment has been performed, and one theory of gravity after another has fallen by the wayside a victim of the observations, Einstein's theory has stood firm. No purported inconsistency between experiment and Einstein's laws of gravity has ever surmounted the test of time.

Query: Why then bother to examine alternative theories of gravity? Reply: To have "foils" against which to test Einstein's theory.
To say that Einstein's geometrodynamics is "battle-tested" is to say it has won every time it has been tried against a theory that makes a different prediction. How then does one select new antagonists for decisive new trials by combat?
Not all theories of gravity are created equal. Very few, among the multitude in the literature, are sufficiently viable to be worth comparison with general relativity or with future experiments. The "worthy" theories are those which satisfy three criteria for viability: self-consistency, completeness, and agreement with past experiment.

Self-consistency is best illustrated by describing several theories that fail this test. The classic example of an internally inconsistent theory is the spin-two field theory of gravity [Fierz and Pauli (1939); Box 7.1 here], which is equivalent to linearized general relativity (Chapter 18). The field equations of the spin-two theory imply that all gravitating bodies move along straight lines in global Lorentz reference frames, whereas the equations of motion of the theory insist that gravity deflects
bodies away from straight-line motion. (When one tries to remedy this inconsistency, one finds oneself being "bootstrapped" up to general relativity; see route 5 of Box 17.2.) Another self-inconsistent theory is that of Kustaanheimo (1966). It predicts zero gravitational redshift when the wave version of light (Maxwell theory) is used, and nonzero redshift when the particle version (photon) is used.

Completeness: To be complete a theory of gravity must be capable of analyzing from "first principles" the outcome of every experiment of interest. It must therefore mesh with and incorporate a consistent set of laws for electromagnetism, quantum mechanics, and all other physics. No theory is complete if it postulates that atomic clocks measure the "interval"

d τ = {(- g_{α β} d x^{α} d x^{β})}^{1 / 2}

constructed from a particular metric. Atomic clocks are complex systems whose behavior must be calculated from the fundamental laws of quantum theory and electromagnetism. No theory is complete if it postulates that planets move on geodesics. Planets are complex systems whose motion must be calculated from fundamental laws for the response of stressed matter to gravity. For further discussion see

§ § 16.4, 20.6

, and 40.9 .

Agreement with past experiment: The necessity that a theory agree, to within several standard deviations, with the "four standard tests" (gravitational redshift, perihelion shift, electromagnetic-wave deflection, and radar time-delay) is obvious. Equally obvious but often forgotten is the need to agree with the expansion of the universe (historically the ace among all aces of general relativity) and with observations at the more everyday, Newtonian level. Example: Birkhoff's (1943) theory predicts the same redshift, perihelion shift, deflection, and time-delay as general relativity. But it requires that the pressure inside gravitating bodies equal the total density of mass-energy,

p = ρ

; and, as a consequence, it demands that sound waves travel with the speed of light. Of course, this prediction disagrees violently with experiment. Therefore, Birkhoff's theory is not viable. Another example: Whitehead's (1922) theory of gravity was long considered a viable alternative to Einstein's theory, because it makes exactly the same prediction as Einstein for the "four standard tests." Not until the work of Will (1971b) was it realized that Whitehead's theory predicts a time-dependence for the ebb and flow of ocean tides that is completely contradicted by everyday experience (see

§ 40.8

§39.2. METRIC THEORIES OF GRAVITY

Two lines of argument narrow attention to a restricted class of gravitation theories, called metric theories.

The first line of argument constitutes the theme of the preceding chapter. It examined experiment after experiment, and reached two conclusions: (1) spacetime possesses a metric; and (2) that metric satisfies the equivalence principle (the standard special relativistic laws of physics are valid in each local Lorentz frame). Theories of gravity that incorporate these two principles are called metric theories.* In brief, Chapter 38 says, "For any adequate description of gravity, look to a metric theory."

Why attention focuses on metric theories of gravity

How metric theories differ

Weak-field, slow-motion expansion of a metric theory

Exception: Cartan's (1922b, 1923) theory ["general relativity plus torsion"; see Trautman (1972)] is nonmetric, but agrees with experiment and is experimentally indistinguishable from general relativity with the technology of the 1970's.

The second line of argument pointing to metric theories begins with the issue of completeness (preceding section). To be complete, a theory must incorporate a self-consistent version of all the nongravitational laws of physics. No one has found a way to incorporate the rest of physics with ease except to introduce a metric, and then invoke the principle of equivalence. Other approaches lead to dismaying complexity, and usually to failure of the theory on one of the three counts of self-consistency, completeness, and agreement with past experiment. All the theories known to be viable in 1973 are metric, except Cartan's. [See Ni (1972b); Will (1972).]

In only one significant way do metric theories of gravity differ from each other: their laws for the generation of the metric. In general relativity theory, the metric is generated directly by the stress-energy of matter and of nongravitational fields. In Dicke-Brans-Jordan theory (Box 39.1, p. 1070), matter and nongravitational fields generate a scalar field

ϕ

; then

ϕ

acts together with the matter and other fields to generate the metric. Expressed in the language of

§ 38.7, ϕ

is a "new long-range field" that couples indirectly to matter. As another example, a theory devised by

Ni (1970, 1972)

(Box 39.1) possesses a flat-space metric

η

and a universal time coordinate

t

("prior geometry"; see §17.6);

η

acts together with matter and nongravitational fields to generate a scalar field

ϕ

; and then

η, t

, and

ϕ

combine to create the physical metric

g

that enters into the equivalence principle.

All three of the above theories-Einstein, Dicke-Brans-Jordan, Ni-were viable in the summer of 1971, when this section was written. But in autumn 1971 Ni's theory, and many other theories that had been regarded as viable, were proved by Nordtvedt and Will (1972) to disagree with experiment. This is an example of the rapidity of current progress in experimental tests of gravitation theory!

Henceforth, in this chapter and the next, attention will be confined to metric theories of gravity and their comparison with experiment.

§39.3. POST-NEWTONIAN LIMIT AND PPN FORMALISM

The solar system, where experiments to distinguish between metric theories are performed, has weak gravity,

\begin{matrix} (39.1a) & | Φ | =∣ Newtonian potential ∣≲ 10^{- 6}; \end{matrix}

moreover, the matter that generates solar-system gravity moves slowly

\begin{matrix} (39.1b) & v^{2} = (velocity relative to solar-system center of mass)^{2} ≲ 10^{- 7} \end{matrix}

and has small stress and internal energies

\begin{matrix} (39.1c) & | T_{j k} | / ρ_{o} = (stress divided by baryon "mass" density) ≦ 10^{- 6}, \\ (39.1d) & Π = (ρ - ρ_{o}) / ρ_{o} = (\binom{internal energy density per}{unit baryon "mass" density}) ≲ 10^{- 6} . \end{matrix}

[Here the baryon "mass" density

ρ_{o}

, despite its name, and despite the fact it is sometimes even more misleadingly called "density of rest mass-energy," is actually a measure of the number density of baryons

n

, and nothing more. It is defined as the product of

n

with some standard figure for the mass per baryon,

μ_{0}

, in some well-defined standard state; thus,

\begin{matrix} (39.1e) & ρ_{o} \equiv n μ_{0} \cdot] \end{matrix}

Consequently, the analysis of solar-system experiments using any metric theory of gravity can be simplified, without significant loss of accuracy, by a simultaneous expansion in the small parameters

| Φ |, v^{2}, | T_{j k} | / ρ_{o}

, and

Π

. Such a "weak-field, slow-motion expansion" gives: (1) flat, empty spacetime in "zero order"; (2) the Newtonian treatment of the solar system in "first order"; and (3) post-Newtonian corrections to the Newtonian treatment in "second order".

The formalism of Newtonian theory plus post-Newtonian corrections is called the "post-Newtonian approximation." Each metric theory has its own post-Newtonian approximation. Despite the great differences between metric theories themselves, their post-Newtonian approximations are very similar. They are so similar, in fact, that one can construct a single post-Newtonian theory of gravity, devoid of any reference to indirectly coupling fields (

ϕ

in Dicke-Brans-Jordan;

η, t

, and

ϕ

in Ni; see Box 39.1), that contains the post-Newtonian approximation of every conceivable metric theory as a special case. This all-inclusive post-Newtonian theory is called the "Parametrized Post-Newtonian (PPN) Formalism." It contains a set of parameters (called "PPN parameters") that can be specified arbitrarily. One set of values for these parameters makes the PPN formalism identical to the post-Newtonian limit of general relativity; another set of values makes it the post-Newtonian limit of Dicke-Brans-Jordan theory, etc.

Subsequent sections of this chapter present a version of the PPN formalism devised by Clifford M. Will and Kenneth Nordtvedt, Jr. (1972). [See also Will (1972).] This version, containing ten PPN parameters, encompasses as special cases nearly every metric theory of gravity known to the authors. The few exceptions [Whitehead (1922) and theories reviewed by Will (1973)] all disagree with experiment. One can include them in the PPN formalism by adding additional terms and parameters.

The ten parameters are described heuristically in Box 39.2, for the convenience of readers who would skip the full details of the formalism (

$ § 39.4 - 39.12

How accurate is the PPN formalism? Or, stated more precisely, how accurately does the post-Newtonian approximation agree with the metric theory from which it comes? In the solar system, where

| Φ |, v^{2}, | T_{j k} | / ρ_{o}

, and

Π

are all

≦ 10^{- 6}

, the post-Newtonian approximation makes fractional errors of

≲ 10^{- 6}

in quantities of post-Newtonian order, and fractional errors of

≲ 10^{- 12}

in quantities of Newtonian order. For example, it misrepresents the deflection of light by

≦ 10^{- 6} \times

(post-Newtonian deflection)

\sim 10^{- 6}

seconds of arc. And it ignores relativistic deformations of the Earth's orbit of magnitude

< 10^{- 12} \times

(one astronomical unit)

\sim 10

centimeters. Clearly, there is no need in the 1970's to use higher-order corrections to the postNewtonian approximation; and hence no need to construct a "parametrized post-post-Newtonian framework." However, in the words of Shapiro (1971b): "If one projects from the achievements in the last decade, it is not unreasonable to predict

Post-Newtonian approximation

PPN formalism

Accuracy of PPN formalism in solar system

Box 39.1 THE THEORIES OF DICKE-BRANS-JORDAN AND OF NI

A. Dicke-Brans-Jordan

References: Brans and Dicke (1961); Jordan (1959). [Notes: This is the special case

η = - 1

of Jordan's theory. An alternative mathematical representation of the theory is given by Dicke (1962).]
Fields associated with gravity:

ϕ

, a long-range scalar field;

g

, the metric of spacetime (from which are constructed the covariant derivative

\nabla

and the curvature tensors, in the usual manner).
Equations by which these fields are determined:
The trace of the stress-energy tensor generates

ϕ

via the curved-spacetime wave equation

◻ ϕ = ϕ_{; α}^{, α} = \frac{8 π}{3 + 2 ω} T,

where

ω

is the dimensionless "Dicke coupling constant."
The stress-energy tensor and

ϕ

together generate the metric (i.e., the spacetime curvature) via the field equations

G_{α β} = \frac{8 π}{ϕ} T_{α β} + \frac{ω}{ϕ^{2}} (ϕ_{, α} ϕ_{, β} - \frac{1}{2} g_{α β} ϕ_{, μ} ϕ^{, μ}) + \frac{1}{ϕ} (ϕ_{; α β} - g_{α β} ◻ ϕ),

where

G_{α β}

is the Einstein tensor.
Variational principle for these equations:

δ \int [ϕ R - ω (ϕ_{, α} ϕ^{, α} / ϕ) + 16 π L] (- g)^{1 / 2} d^{4} x = 0

where

R

is the scalar curvature and

L

is the matter Lagrangian.
Equivalence principle is satisfied:
The special-relativistic laws of physics are valid, without change, in the local Lorentz frames of the metric

g

.
Consequence: the scalar field does not exert any direct influence on matter; its only role is that of participant in the field equations that determine the geometry of spacetime. It is an "indirectly coupling field" in the sense of §38.7.
This theory is self-consistent, complete, and for

ω > 5

in "reasonable" accord (two standard deviations or better) with all pre-1973 experiments.

B. Ni

References: Ni

(1970, 1972)

Fields associated with gravity:

η

, a flat "background metric" ("prior geometry" in sense of §17.6). There exist,
by assumption, coordinate systems ("background Lorentz frames") in which everywhere at once

η_{00} = - 1, η_{0 j} = 0

, and

η_{j k} = δ_{j k}

t

, a scalar field called the "universal time coordinate" ("prior geometry" in sense of §17.6), which is so "tuned" to the background metric that

t_{1 α β} = 0, t_{, α}^{t},_{β} η^{α β} = - 1

where "" denotes covariant derivative with respect to

η

.
This means there exists a background Lorentz frame (the "rest frame of the universe") in which

x^{0} = t

ϕ

, a scalar field called the "scalar gravitational field".

g

, the metric of spacetime (from which are constructed the covariant derivative

\nabla

and the curvature tensors, in the usual manner).
Equations by which these fields are determined:
The stress-energy of spacetime generates the scalar gravitational field

ϕ

via the wave equation

\begin{aligned} ◻ ϕ & \equiv ϕ^{, α}; α = - 2 π T^{α β} \partial g_{α β} / \partial ϕ \\ = 4 π T^{α β} [η_{α β} e^{- 2 ϕ} + (e^{2 ϕ} + e^{- 2 ϕ}) t_{, α} t_{, β}] . \end{aligned}

ϕ, η

, and

t

together determine the metric of spacetime through the algebraic relation

g = e^{- 2 ϕ} η + (e^{- 2 ϕ} - e^{2 ϕ}) d t \otimes d t .

Note: In the "rest frame of the universe" that is presupposed in this theory, this metric reduces to

d s^{2} = g_{α β} d x^{α} d x^{β} = - e^{2 ϕ} d t^{2} + e^{- 2 ϕ} (d x^{2} + d y^{2} + d z^{2})

Variational principle for the field equation for

ϕ

:
where

L

is the matter Lagrangian.
Equivalence principle is satisfied:
The special-relativistic laws of physics are valid, without change, in the local Lorentz frames of the spacetime metric

g

.
Consequence:

ϕ, η

, and

t

do not exert any direct influence on matter; they are "indirectly coupling fields" in the sense of

$ 38.7

.
This theory is self-consistent and complete. If the solar system were at rest in the
"rest frame of the universe", the theory would agree with all experiments to date-except, possibly, the expansion of the universe. But the motion of the solar system through the universe leads to serious disagreement with experiment (Will and Nordtvedt 1972; §40.8).

Box 39.2 HEURISTIC DESCRIPTION OF THE TEN PPN PARAMETERS

Parameter	What it measures, relative to general relativity $^{a}$	Value in General Relativity	Value in Dicke-BransJordan Theory $^{b}$	Value in Ni's Theory $^{b}$
$γ$	How much space curvature $(g_{j k})$ is produced by unit rest mass?	1	$\frac{1 + ω}{2 + ω}$	1
$β$	How much nonlinearity is there in the superposition law for gravity $(g_{00})$ ?	1	1	1
$β_{1}$	How much gravity ( $g_{00}$ ) is produced by unit kinetic energy $(\frac{1}{2} ρ_{o} v^{2})$ ?	1	$\frac{3 + 2 ω}{4 + 2 ω}$	1
$β_{2}$	How much gravity ( $g_{00}$ ) is produced by unit gravitational potential energy $(ρ_{o} U)$ ?	1	$\frac{1 + 2 ω}{4 + 2 ω}$	1
$β_{3}$	How much gravity ( $g_{00}$ ) is produced by unit internal energy ( $ρ_{0} Π$ ) ?	1	1	1
$β_{4}$	How much gravity ( $g_{00}$ ) is produced by unit pressure (p)?	1	$\frac{1 + ω}{2 + ω}$	1
$ζ$	How much more gravity ( $g_{00}$ ) is produced by radial kinetic energy $[\frac{1}{2} ρ_{o} (v \cdot \hat{r})^{2}]$ -i.e., kinetic energy of motion toward obser-ver-than by transverse kinetic energy?	0	0	0
$η$	How much more gravity $(g_{00})$ is produced by radial stress $[\hat{r} \cdot t \cdot \hat{r}]$ than by transverse stress?	0	0	0
$Δ_{1}$	How much dragging of inertial frames $(g_{0 j})$ is produced by unit momentum $(ρ_{o} v)$ ?	1	$\frac{10 + 7 ω}{14 + 7 ω}$	$- \frac{1}{7}$
$Δ_{2}$	How much easier is it for momentum ( $ρ_{o} v$ ) to drag inertial frames radially (toward the observer) than in a transverse direction?	1	1	1

that in the 1980's techniques will be available to detect second-order effects of general relativity. At that point the ratio of theoretical to experimental relativists may take a sharp turn downwards."

Actually, there are a few exceptions to the claim that the post-Newtonian approximation suffices for the 1970's. These exceptions occur where the external universe impinges on and influences the solar system. For example, gravitational waves propagating into the solar system from distant sources (Chapters 35-37) are ignored by every post-Newtonian approximation and by the PPN framework. They must be treated using a full metric theory or a weak-field, "fast-motion" approximation
to such a theory. Similarly, time-dependence of the "gravitational constant" (

$ 40.8

), induced in some theories by expansion of the universe, is beyond the scope of the PPN formalism, as is the expansion itself.

The PPN formalism is used not only in interpreting experimental tests of gravitation theories, but also as a powerful tool in theoretical astrophysics. By specializing all the PPN parameters to unity, except

ζ = η = 0

, one obtains the post-Newtonian approximation to Einstein's theory of gravity. This post-Newtonian approximation can then be used (and has been used extensively) to calculate general relativistic corrections to such phenomena as the structure and stability of stars.*

Historical and Notational Notes

The earliest parametrizations of the post-Newtonian approximation were performed, and used in interpreting solar system experiments, by Eddington (1922), Robertson (1962), and Schiff

(1962, 1967)

. However, they dealt solely with the vacuum gravitational field outside an isolated, spherical body (the sun). Nordtvedt (1968b, 1969) devised the first full PPN formalism, capable of treating all aspects of the solar system; he treated the sun, planets, and moon as made from "gases" of point-particles (atoms) that interact gravitationally and electromagnetically. Will (1971c) later used techniques devised by Chandrasekhar (1965a) to modify Nordtvedt's formalism, so that it employs a stressed, continuous-matter description of celestial bodies. The version of the formalism presented here, devised by Will and Nordtvedt (1972), generalizes all previous versions to acquire "post-Galilean invariance" [see Chandrasekhar and Contopolous (1967)]. The most detailed and up-to-date review article on the PPN formalism is Will (1972).

In the literature of post-Newtonian physics and the PPN formalism, the Newtonian potential is described traditionally not by

Φ

, but by

\begin{matrix} (39.2) & U \equiv - Φ \equiv + \int \frac{ρ_{o} (x^{'}) d^{3} x^{'}}{| x - x^{'} |} \end{matrix}

To avoid confusion, this chapter and the next will use

U

, although the rest of the book uses

Φ

Turn now to a detailed, Track-2 exposition of the PPN formalism.

§39.4. PPN COORDINATE SYSTEM

The PPN formalism covers the solar system (or whatever system is being analyzed) with coordinates

(t, x_{j}) \equiv (t, x^{j})

that are as nearly globally Lorentz as possible:

\begin{matrix} (39.3) & g_{α β} = η_{α β} + h_{α β}, | h_{α β} | ≦ M_{⊙} / R_{⊙} \sim 10^{- 6} . \end{matrix}

Applications of PPN formalism to astrophysics

History and notation of PPN formalism

EXPOSITION OF PPN
FORMALISM:
Coordinate system

The rest of this chapter is Track 2. No earlier Track-2 material is needed as preparation for it, but the following will be helpful:
(1) Chapter 7 (incompatibility of gravity and special relativity)
(2) § 17.6 (no prior geometry);
(3) §§36.9-36.11 (generation of gravitational waves); and
(4) Chapter 38 (tests of foundations).
This chapter is not needed as preparation for any later chapter, but it will be helpful in Chapter 40 (solar-system tests)
(In this sense the PPN formalism is like linearized theory; see Chapter 18.) The velocity of the coordinate system (i.e., 4 -velocity of its spatial origin) is so chosen that the solar system is approximately at rest in these coordinates. (Whether the center of mass of the solar system is precisely at rest, or is moving with some low velocity

v ≦ {(M_{⊙} / R_{⊙})}^{1 / 2} \sim 10^{- 3} \sim 300 km / \sec

, is a matter for the user of the formalism to decide. For more on the options, see

§ § 39.9

and 39.12.)

The PPN coordinates provide one with a natural "

3 + 1

" split of spacetime into space plus time. That split is conveniently treated using the notation of three-dimensional, flat-space vector analysis-even though spacetime and the three-dimensional hypersurfaces

x^{0} =

constant are both curved. The resultant three-dimensional formalism will look more like Newtonian theory than like general relativity-as, indeed, one wishes it to; after all, one's goal is to study small relativistic corrections to Newtonian theory!

§39.5. DESCRIPTION OF THE MATTER IN THE SOLAR SYSTEM

Description of matter

Relative to the PPN coordinates, the matter of the solar system (idealized as a stressed medium) has a coordinate-velocity field

\begin{matrix} (39.4) & v_{j} \equiv d x_{j} / d x^{0} \end{matrix}

Choose an event

P

, and in its neighborhood transform to an orthonormal frame that moves with the matter there. Orient the spatial axes

e_{j}

of this comoving frame so that they coincide as accurately as possible with the PPN coordinate axes. (This requirement will be made more precise in §39.10.) In the orthonormal comoving frame, define the following quantities, which describe the state of the matter:

\begin{aligned} (39.5a) & (density of total mass-energy) \equiv ρ; \\ (baryon "mass" density) \equiv ρ_{o} \\ (39.5b) & \equiv (\binom{number density}{of baryons, n}) \times (\binom{standard rest mass per baryon, μ_{0}}{for matter in some standard state}); \\ (39.5c) & (specific internal energy density) \equiv Π \equiv (ρ - ρ_{o}) / ρ_{o}; \\ (39.5d) & (components of stress tensor) \equiv t_{\hat{i} \hat{j}} \equiv e_{\hat{i}} \cdot T \cdot e_{\hat{j}}; \\ (39.5e) & \begin{array}{r} (pressure) \equiv p \equiv \frac{1}{3} (t_{\hat{x} \hat{x}} + t_{\hat{y} \hat{y}} + t_{\hat{z} \hat{z}}) \\ \equiv (average of stress over all directions). \end{array} \end{aligned}

Anisotropies (i.e., shears) in the stress are important only in planets such as the Earth; and even there they are dominated by the isotropic pressure:

\begin{matrix} (39.6) & t_{i j} = p δ_{i j} + p \times [corrections ≪ 1] . \end{matrix}

For many purposes, especially inside the sun, one can ignore the anisotropies, thereby approximating the solar-system matter as a perfect fluid.*

The isotropic part of the radiation field gives a significant contribution to the pressure,

p

, and the density of internal energy,

ρ_{o} Π

, inside the sun. However, the anisotropic radiation flux is ignored in the stress-energy tensor. This approximation is allowable because in the sun the outward energy flux carried by radiation is less than

10^{- 15}

of the internal energy density

ρ_{o} Π

; in planets it is even less.

§39.6. NATURE OF THE POST-NEWTONIAN EXPANSION

For any gravitationally bound configuration such as the solar system, the Newtonian approximation imposes limits on the sizes of various dimensionless physical quantities (see exercise 39.1):

\begin{aligned} (39.7) & ϵ^{2} \equiv & maximum value of Newtonian potential U \\ ≳ values anywhere of U, v^{2}, p / ρ_{o}, | t_{i j} | / ρ_{o}, Π . \end{aligned}

Relative magnitudes of expansion parameters
(The Newtonian potential at the center of the sun is

ϵ^{2} \sim 10^{- 5}

. The values of

p / ρ_{o}

t_{i j} / ρ_{o}

, and

Π

there are also

\sim 10^{- 5}

, and they are much smaller elsewhere. The orbital velocities of the planets are all less than

100 km / \sec = 3 \times 10^{- 4}

, so

v^{2} < 10^{- 7}

.) Moreover, changes with time of all quantities at fixed

x_{j}

are due primarily to the motion of the matter. As a result, time derivatives are small by

O (ϵ)

compared to space derivatives,

\begin{matrix} (39.8) & | \frac{\partial A / \partial t}{\partial A / \partial x_{j}} | \sim | v_{j} | \leq ϵ for any quantity A, \end{matrix}

although not in the radiation zone, where outgoing gravitational waves flow (distance

\geq

one light year from Sun). Consequently, the radiation zone must be excluded from the analysis when one makes a post-Newtonian expansion. To treat it requires different techniques, e.g., those of Chapter 36.

Conditions 39.7 and 39.8 suggest that one expand the metric coefficients in powers of the small parameter

ϵ

, treating

U, v^{2}, p / ρ_{o}, t_{i j} / ρ_{o}

, and

Π

as though they were all of

O (ϵ^{2})

(often they are smaller!), and treating time derivatives as

O (ϵ)

smaller than space derivatives.

In this "post-Newtonian" expansion, terms odd in

ϵ

(i.e., terms such as

\begin{matrix} (39.9) & \int \frac{ρ_{o} (x^{'}, t) v_{j} (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'} \sim \frac{M}{R} v \sim ϵ^{3} \end{matrix}

whose total number of

v

's and

(\partial / \partial t)

's is odd) change sign under time reversal,

whereas terms even in

ϵ

do not. Time reversal

(x^{\overset{―}{0}} = - x^{0})

also changes the sign of

g_{0 j} (g_{\overset{―}{0} j} = - g_{0 j})

, but leaves

g_{00}

and

g_{j k}

unchanged. Therefore,

g_{0 j}

must contain only terms odd in

ϵ

; whereas

g_{00}

and

g_{j k}

must contain only even terms. (Actually, this ceases to be the case when radiation damping enters the picture. In the real world one always insists on outgoing-wave boundary conditions. But time reversal converts outgoing waves to ingoing waves; so an extra sign change is required to convert back to out. Therefore, radiation damping terms in the near-zone metric are even in

ϵ

for

g_{0 j}

, but odd for

g_{00}

and

g_{j k}

. However, radiation damping does not come into play until order

ϵ^{5}

beyond Newtonian theory-see Chapter 36-so it will be ignored here.)
Expanded form of metric
The form of the expansion is already known through Newtonian order (see §17.4, with

Φ

replaced by

- U

): Newtonian gravity is only obtained when one demanc. that
$$

\begin{aligned} g_{00} & = - 1 + 2 U + [terms ≲ ϵ^{4}], \\ (39.10) & g_{0 j} & = [terms ≲ ϵ^{3}], \\ g_{i j} & = δ_{i j} + [terms \leq ϵ^{2}] . \end{aligned}

The stated limits on the higher-order corrections are dictated by demanding that the space components of the geodesic equation agree with the Newtonian equation of motion:

\begin{aligned} (39.11) & \frac{d^{2} x_{j}}{d t^{2}} \approx \frac{d^{2} x_{j}}{d τ^{2}} & = - Γ^{j}_{α β} \frac{d x^{α}}{d τ} \frac{d x^{β}}{d τ} \approx - Γ^{j}_{α β} \frac{d x^{α}}{d t} \frac{d x^{β}}{d t} \\ = - Γ^{j}_{00} - 2 Γ^{j}_{o k} v_{k} - Γ^{j}_{k!} v_{k} v_{ℓ} \\ = U_{, j} + terms of order {ϵ g_{0 k, j}; ϵ^{2} g_{k ℓ, j}} . \end{aligned}

One would get the wrong Newtonian limit if

g_{0 k}

were

O (ϵ)

or greater, and if

g_{k ℓ} - δ_{k ι}

were

O (1)

or greater.
The above pattern continues on to all orders in the expansion. Thus in the geodesic equation, and also in the law of local conservation of energy-momentum

T^{α β}_{; β} = 0

g_{00}

always goes hand-in-hand with

ϵ g_{0 k}

and

ϵ^{2} g_{j k}

(see exercise 39.2). Therefore, the post-Newtonian expansion has the form summarized in Box 39.3.

EXERCISES

Exercise 39.1. ORDERS OF MAGNITUDE IN GRAVITATIONALLY BOUND SYSTEMS

Use Newtonian theory to derive conditions (39.7) for any gravitationally bound system. [Hint: Such concepts as orbital velocities, the speeds of sound and shear waves, the virial theorem, and hydrostatic equilibrium are relevant.]

Exercise 39.2. PATTERN OF TERMS IN POST-NEWTONIAN EXPANSION

Verify the statements in the paragraph following equation (39.11). In particular, suppose that one wishes to evaluate the coordinate acceleration,

d^{2} x_{j} / d t^{2}

, to accuracy

ϵ^{2 N} U_{, j}

for some
integer

N

. Show that this undertaking requires a knowledge of

g_{00}

to accuracy

ϵ^{2 N + 2}

, of

g_{0 k}

ϵ^{2 N + 1}

, and of

g_{i k}

ϵ^{2 N}

. Also suppose that one knows

T^{00}

to accuracy

ρ_{o} ϵ^{2 N}, T^{0 j}

ρ_{o} ϵ^{2 N + 1}

, and

T^{j k}

ρ_{o} ϵ^{2 N + 2}

[see, e.g., equations (39.13) for

N = 0

and (39.42) for

N = 2

]. Show that to calculate

T^{0 α}_{; α}

with accuracy

ϵ^{2 N + 1} ρ_{o, j}

and

T^{j α}_{; α}

with accuracy

ϵ^{2 N + 2} ρ_{o, j}

, one must know

g_{00}

ϵ^{2 N + 2}, g_{0 k}

ϵ^{2 N + 1}

, and

g_{j k}

ϵ^{2 N}

. This dictates the pattern of Box 39.3.

§39.7. NEWTONIAN APPROXIMATION

At Newtonian order the metric has the form (39.10); and the 4-velocity and stress- Newtonian approximation energy tensor have components, relative to the PPN coordinate system,

\begin{aligned} (39.12) & u^{0} & = + 1 + O (ϵ^{2}), u^{j} = v_{j} + O (ϵ^{3}) \\ T^{00} & = ρ_{o} + O (ρ_{o} ϵ^{2}), T^{0 j} = ρ_{o} v_{j} + O (ρ_{o} ϵ^{3}), \\ (39.13) & T^{j k} & = t_{j \hat{j} k} + ρ_{o} v_{j} v_{k} + O (ρ_{o} ϵ^{4}) \end{aligned}

(see exercise 39.3). Two sets of equations govern the structure and evolution of the solar system. (1) The Einstein field equations. As was shown in §18.4, and also in

§ 17.4

, in the Newtonian limit Einstein's equations reduce to Laplace's equation

\begin{matrix} (39.14a) & U_{, j j} = - 4 π ρ_{o} \end{matrix}

which has the "action-at-a-distance" solution

\begin{matrix} (39.14b) & U (x, t) = \int \frac{ρ_{o} (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'} \end{matrix}

Box 39.3 POST-NEWTONIAN EXPANSION OF THE METRIC COEFFICIENTS

Level of approximation

(and papers expanding general

relativity to this level)

Order or value of terms

flat, empty spacetime

-1

g_{00}

Newtonian approximation

2 U

g_{j k}

post-Newtonian approximation

[(Fock (1959); Chandrasekhar (1965a)]

+ terms

\sim ϵ^{4}

+ terms

\sim ϵ^{3}

+ terms

\sim ϵ^{2}

post-post-Newtonian approximation

[Chandrasekhar and Nutku (1969)]

+ terms

\sim ϵ^{6}

+ terms

\sim ϵ^{5}

+ terms

\sim ϵ^{4}

radiation damping

[Chandrasekhar and Esposito (1970)]

+ terms

\sim ϵ^{7}

+ terms

\sim ϵ^{6}

+ terms

\sim ϵ^{5}

(2) The law of local energy-momentum conservation,

T^{α β}_{; β} = 0

. The time component of this law reduces to the conservation of rest mass

\begin{matrix} (39.15a) & \partial ρ_{o} / \partial t + \partial (ρ_{o} v_{j}) / \partial x_{j} = 0 + fractional errors of O (ϵ^{2}); \end{matrix}

and the space components reduce to Newton's second law of motion, "

F = m a

\begin{matrix} (39.15b) & ρ_{o} d v_{j} / d t = ρ_{o} (\partial U / \partial x_{j}) - \partial t_{j k} / \partial x_{k} + fractional errors of O (ϵ^{2}), \\ (39.16) & d / d t \equiv (time derivative following the matter) \equiv \partial / \partial t + v_{k} \partial / \partial x_{k} \end{matrix}

(see exercise 39.3).
Equations (39.14)-(39.16), together with equations of state describing the planetary and solar matter, are the foundations for all Newtonian calculations of the structure and motion of the sun and planets. Notice that the internal energy density

ρ_{0} Π

nowhere enters into these equations. It is of no importance to Newtonian hydrodynamics. It matters for the sun's thermal-energy balance; but that is irrelevant here.

EXERCISES

Exercise 39.3. NEWTONIAN APPROXIMATION

(a) Derive equations (39.13) for the components of the stress-energy tensor in the PPN coordinate frame. [Hint: In the rest frame of the matter ("comoving orthonormal frame")

T_{\hat{0} \hat{0}} = ρ = ρ_{o} + O (ϵ^{2}), T_{\hat{0} \hat{j}} = 0, T_{\hat{j} \hat{k}} = t_{\hat{j} k}

; see equations (39.5). Lorentz-transform these components by a pure boost with ordinary velocity

- v_{j}

to obtain

T_{α β}

.]
(b) Show that, in the PPN coordinate frame,

T^{0 α}_{; α} = 0

reduces to equation (39.15a), and

T_{; α}^{j α} = 0

, when combined with (39.15a), reduces to equation (39.15b).]

Exercise 39.4. A USEFUL FORMULA

Derive from equations (39.15) the following useful formula, valid for any function

f (x, t)

\begin{aligned} (39.17) & \frac{d}{d t} \int ρ_{o} (x, t) f (x, t) d^{3} x = & \int ρ_{o} (x, t) \frac{d f (x, t)}{d t} d^{3} x \\ + fractional errors of O (ϵ^{2}) . \end{aligned}

Here both integrals are extended over all of space; and

d f / d t

is the derivative following the matter (39.16).

Exercise 39.5. STRESS TENSOR FOR NEWTONIAN GRAVITATIONAL FIELD

Define a "stress tensor for the Newtonian gravitational field

U

" as follows:

\begin{matrix} (39.18) & t_{j k} \equiv \frac{1}{4 π} (U_{, j} U_{, k} - \frac{1}{2} δ_{j k} U_{, l} U_{, l}) . \end{matrix}

Show that the equations of motion for the matter (39.15b) can be rewritten in the forms

\begin{matrix} (39.19) & ρ_{o} \frac{d v_{j}}{d t} = - \frac{\partial}{\partial x^{k}} (t_{j k} + t_{j k}) + fractional errors of O (ϵ^{2}), \\ (39.19') & {(ρ_{o} v_{j})}_{, t} + {(t_{j k} + t_{j k} + ρ_{o} v_{j} v_{k})}_{, k} = 0 + fractional errors of O (ϵ^{2}) . \end{matrix}

Exercise 39.6. NEWTONIAN VIRIAL THEOREMS

(a) From equation

({39.19}^{'})

show that

\begin{matrix} (39.20a) & d^{2} I_{j k} / d t^{2} = 2 \int (t_{j k} + t_{j k} + ρ_{o} v_{j} v_{k}) d^{3} x + fractional errors of O (ϵ^{2}) \end{matrix}

where

I_{j k}

is the second moment of the system's mass distribution,

I_{j k} = \int ρ_{o} x_{j} x_{k} d^{3} x

This is called the "time-dependent tensor virial theorem."
(b) From this infer that, if

⟨ ⟩_{longtime}

denotes an average over a long period of time, then

\begin{matrix} (39.20b) & {⟨ \int (t_{j k} + t_{3 \hat{k}} + ρ_{o} v_{j} v_{k}) d^{3} x ⟩}_{long time} = O (\int ρ_{o} ϵ^{4} d^{3} x) . \end{matrix}

This is called the "tensor virial theorem."
(c) By contraction of indices and use of equations (39.18), (39.14a), and (39.5e), derive the (ordinary) virial theorems:

\frac{1}{2} d^{2} I / d t^{2} = \int ρ_{o} v^{2} d^{3} x - \int \frac{1}{2} ρ_{o} U d^{3} x + 3 \int p d^{3} x +

fractional errors of

O (ϵ^{2})

,
where

I

is the trace of the second moment of the mass distribution

I = I_{j j} = \int ρ_{o} r^{2} d^{3} x

and

\begin{aligned} (39.2lb) & ⟨ \underset{⏟}{\int ρ_{0} v^{2} d^{3} x} - \underset{kinetic}{\underset{⏟}{\frac{1}{2} \int ρ_{0} U d^{3} x}} + \underset{(\binom{gravitational}{energy}) + 3 \times (\binom{pressure}{integral})}{\underset{⏟}{3 \int p d^{3} x}} ⟩_{longtime} = O (\int ρ_{0} ϵ^{4} d^{3} x) \\ 2 \times ( \end{aligned}

Exercise 39.7. PULSATION FREQUENCY FOR NEWTONIAN STAR

Use the ordinary, time-dependent virial theorem (39.21a) to derive the following equation for the fundamental angular frequency of pulsation of a nonrotating, Newtonian star:

\begin{matrix} (39.22a) & ω^{2} = (3 {\bar{Γ}}_{1} - 4) \frac{∣ star's self-gravitational energy ∣}{(trace of second moment of star's mass distribution)} \\ (39.22b) & {\bar{Γ}}_{1} = (\binom{pressure-weighted average}{of adiabatic index}) \equiv \frac{\int Γ_{1} p d^{3} x}{\int p d^{3} x} \end{matrix}

In the derivation assume that the pulsations are "homologous"-i.e., that a fluid element with equilibrium position

x^{j}

(relative to center of mass

x^{j} = 0

) gets displaced to

x^{j} + ξ^{j} (x, t)

, where

ξ^{j} = (small constant) x^{j} e^{- i ω t} .

Assume nothing else. Notes: (1) The result (39.22) was derived differently in Box 26.2 and used in

$ 24.4

. (2) The assumption of homologous pulsation is fully justified if

| Γ_{1} - 4 / 3 | =

constant

≪ 1

; see Box 26.2 . (3) The result (39.22) is readily generalized to slowly
rotating Newtonian stars; see, e.g., Chandrasekhar and Lebovitz (1968). It can also be generalized to nonrotating post-Newtonian stars using general relativity (Box 26.2), or using the PPN formalism for any metric theory [ Ni (1973)]. And it can be generalized to slowly rotating, post-Newtonian stars [see, e.g., Chandrasekhar and Lebovitz (1968)].

§39.8. PPN METRIC COEFFICIENTS

Post-Newtonian corrections to metric:
(1) rules governing forms
(2) construction of corrections

The post-Newtonian corrections

k_{α β}

to the metric coefficients

g_{α β}

are calculated, in any metric theory of gravity, by lengthy manipulations of the field equations. (See, e.g., exercise 39.14 near the end of this chapter for general relativity.) But without ever picking some one theory, and without ever writing down any set of field equations, one can infer the forms of the post-Newtonian corrections

k_{α β}

. Their forms are fixed by the following constraints: (1) They must be of post-Newtonian order (

k_{00} \sim ϵ^{4}, k_{0 j} \sim ϵ^{3}, k_{i j} \sim ϵ^{2}

). (2) They must be dimensionless. (3)

k_{00}

must be a scalar under rotations,

k_{0 j}

must be components of a 3 -vector, and

k_{j k}

must be components of a 3-tensor. (4) The corrections must die out at least as fast as

1 / r

far from the solar system, so that the coordinates become globally Lorentz and spacetime becomes flat at

r = \infty

. (5) For simplicity, one can assume that the metric components are generated only by

ρ_{o}, ρ_{o} Π, t_{i \hat{j}}, p

, products of these with the velocity

v_{j}

, and time-derivatives of such quantities;* but not by their spatial gradients. [This assumption of simplicity is satisfied by all metric theories examined up to 1973, except Whitehead (1922) and theories reviewed by Will (1973)—which disagree with experiment.] Note the further justification for this assumption in exercise 39.8.

Begin with the corrections to the spatial components,

k_{i j} \sim ϵ^{2}

. There are only two functionals of

ρ_{o}, p, Π, t_{j k}, v_{j}

, that die out at least as fast as

1 / r

, are dimensionless, are

O (ϵ^{2})

, and are second-rank, symmetric 3-tensors; they are

\begin{matrix} (39.23a) & δ_{i j} U (x, t); U_{i j} (x, t) = \int \frac{ρ_{o} (x^{'}, t) (x_{i} - x_{i}^{'}) (x_{j} - x_{j}^{'})}{{| x - x^{'} |}^{3}} d^{3} x^{'} . \end{matrix}

Thus,

k_{i j}

must be

k_{i j} = 2 γ δ_{i j} U + 2 Γ U_{i j}

, for some constant "PPN parameters"

γ

and

Γ

. By an infinitesimal coordinate transformation

[x_{i NEW} = x_{i OLD} + Γ \partial χ / \partial x_{i}

, with

χ (x, t) = - \int ρ_{o} (x^{'}, t) | x - x^{'} | d^{3} x^{'}]

one can set

Γ = 0

, thereby obtaining

\begin{matrix} (39.23b) & g_{i j} = δ_{i j} + k_{i j} = δ_{i j} (1 + 2 γ U) + O (ϵ^{4}) \end{matrix}

*One allows for time derivatives because retarded integrals contain such terms when expanded to post-Newtonian order; thus,

\int \frac{ρ_{o} (x^{'}, t - | x - x^{'} |)}{| x - x^{'} |} d^{3} x^{'} = \int [\frac{ρ_{o} (x^{'}, t)}{| x - x^{'} |} - \frac{\partial ρ_{o} (x^{'}, t)}{\partial t} + \dots] d^{3} x^{'}

However, it turns out that, with a suitable choice of coordinates ("gauge"), all time-odd retarded terms [e.g.,

\int (\partial ρ_{o} / \partial t) d^{3} x

] vanish, except at "the post

^{5 / 2}

-Newtonian order" and at higher orders of approximation; there they lead to radiation damping (see Box 39.3). For example,

\int (\partial ρ_{o} / \partial t) d^{3} x = (d / d t) \int ρ_{o} d^{3} x

vanishes by virtue of the conservation of baryon number.

Next consider

k_{0 j} \sim ϵ^{3}

. Trial and error yield only two vector functionals that die out as

1 / r

or faster, are dimensionless, and are

O (ϵ^{3})

. They are

\begin{matrix} (39.23c) & V_{j} (x, t) = \int \frac{ρ_{o} (x^{'}, t) v_{j} (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'}, \\ (39.23d) & W_{j} (x, t) = \int \frac{ρ_{o} (x^{'}, t) [(x - x^{'}) \cdot v (x^{'}, t)] (x_{j} - x_{j}^{'}) d^{3} x^{'}}{{| x - x^{'} |}^{3}} \end{matrix}

Thus,

k_{0 j}

must be a linear combination of these, involving unknown constants (PPN parameters)

Δ_{1}

and

Δ_{2}

\begin{matrix} (39.23e) & g_{0 j} = k_{0 j} = - \frac{7}{2} Δ_{1} V_{j} - \frac{1}{2} Δ_{2} W_{j} + O (ϵ^{5}) \end{matrix}

Finally consider

k_{00} \sim ϵ^{4}

. Trial and error yields a variety of terms, which can all be combined together with the Newtonian part of

g_{00}

to give

\begin{matrix} (39.23f) & g_{00} = - 1 + 2 U + k_{00} = - 1 + 2 U - 2 β U^{2} + 4 Ψ - ζ a - η D, \end{matrix}

where

\begin{matrix} (39.23~g) & Ψ (x, t) = \int \frac{ρ_{o} (x^{'}, t) ψ (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'}, \\ ψ = β_{1} v^{2} + β_{2} U + \frac{1}{2} β_{3} Π + \frac{3}{2} β_{4} p / ρ_{o}, \\ (39.23h) & a (x, t) = \int \frac{ρ_{o} (x^{'}, t) {[(x - x^{'}) \cdot v (x^{'}, t)]}^{2}}{{| x - x^{'} |}^{3}} d^{3} x^{'}, \\ (39.23i) & D (x, t) = \int \frac{[t_{3 k} (x^{'}, t) - \frac{1}{3} δ_{j k} t_{ℓ ℓ} (x^{'}, t)] (x_{j} - x_{j}^{'}) (x_{k} - x_{k})}{{| x - x^{'} |}^{3}} d^{3} x^{'} . \end{matrix}

Also,

β, β_{1}, β_{2}, β_{3}, β_{4}, ζ, η

are unknown constants (PPN parameters). Elsewhere in the literature the term

- η^{Q}

g_{00}

is ignored (see footnote on p . 1075).

Yet another term is possible: one could have set

\begin{aligned} g_{00} = & [value in equation (39.23 f)] \\ (39.24) & - Σ \iint \frac{ρ_{o} (x^{'}, t) ρ_{o} (x^{''}, t) [(x - x^{'}) \cdot (x^{'} - x^{''})]}{| x - x^{'} | {| x^{'} - x^{''} |}^{3}} d^{3} x^{'} d^{3} x^{''} \end{aligned}

where

Σ

is another PPN parameter. [It can be shown, using the Newtonian equations (39.14)-(39.16), that this expression dies out as

1 / r

far from the solar system.] If

Rigidity of coordinate system

Summary of PPN metric and parameters
such a

Σ

term had been included, then one could have removed it by making the infinitesimal coordinate transformation

\begin{matrix} (39.25) & x_{new}^{0} = x_{old}^{0} - \frac{1}{2} Σ \int \frac{ρ_{o} (x^{'}, t) [(x - x^{'}) \cdot v (x^{'}, t)]}{| x - x^{'} |} d^{3} x^{'} \end{matrix}

(see exercise 39.9). Thus, there is no necessity to include the

Σ

term.
The absence of the

Σ

term from

g_{00}

means that the time coordinate has been fixed rigidly up through post-Newtonian order:

\begin{matrix} (39.26a) & x^{0} has uncertainties only of O (R_{⊙} ϵ^{5}) \sim 10^{- 14} seconds. \end{matrix}

The space coordinates are also fixed rigidly through post-Newtonian order:

\begin{matrix} (39.26b) & x^{j} has uncertainties only of O (R_{⊙} ϵ^{4}) \sim 0.1 cm, \end{matrix}

because any transformation of the form

x^{j}_{new} = x^{j}_{old} + position-dependent terms of O (ϵ^{2} R_{⊙})

would destroy the form ( 39.23 b ) of the space part of the metric.
In summary, for almost every metric theory of gravity yet invented, accurate through post-Newtonian order the metric coefficients have the form (39.23). One theory is distinguished from another by the values of its ten "post-Newtonian parameters"

β, β_{1}, β_{2}, β_{3}, β_{4}, γ, ζ, η, Δ_{1}

and

Δ_{2}

. These are determined by comparing the field equations of the given theory with the form (39.23) of the post-Newtonian metric. The parameter values for general relativity and for several other theories are given in Box 39.2 , along with a heuristic description of each parameter.

EXERCISES

Exercise 39.8. ABSENCE OF "METRIC-GENERATES-METRIC" TERMS IN POST-NEWTONIAN LIMIT

In writing down the post-Newtonian metric corrections, one might be tempted to include terms that are generated by the Newtonian potential acting alone, without any direct aid from the matter. After all, general relativity and other metric theories are nonlinear; so the two-step process (matter)

⟶ U ⟶

(post-Newtonian metric corrections) seems quite natural. Show that such terms are not needed, because the equations (39.14)-(39.16) of the Newtonian approximation enable one to reexpress them in terms of direct integrals over the matter distribution. In particular, show that

\begin{matrix} (39.27) & \int \frac{\partial^{2} U (x^{'}, t) / \partial x_{j}^{'} \partial t}{| x - x^{'} |} d^{3} x^{'} = 2 π [V_{j} (x, t) - W_{j} (x, t)] \end{matrix}

where

V_{j}

and

W_{j}

are defined by equations (

39.23 c, d

); also show that

\begin{aligned} \int \frac{[\partial U (x^{'}, t) / \partial x_{j}^{'}] [\partial U (x^{'}, t) / \partial x_{j}^{'}]}{| x - x^{'} |} d^{3} x^{'} \\ (39.28) & = - 2 π [U (x, t)]^{2} + 4 π \int \frac{ρ_{o} (x^{'}, t) U (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'} . \end{aligned}

Note that the terms on the righthand sides of (39.27) and (39.28) are already included in the expressions (

39.23 e, f

) for

g_{0 j}

and

g_{00}

Exercise 39.9. REMOVAL OF $Σ$ TERM FROM $g_{00}$

Show that the coordinate transformation (39.25) removes the

Σ

term from the metric coefficient

g_{00}

of equation (39.24), as claimed in the text.

Exercise 39.10. VERIFICATION OF FORMS OF POST-NEWTONIAN CORRECTIONS
Verify the claims in the text immediately preceding equations (

39.23 a, b, c, f

§39.9. VELOCITY OF PPN COORDINATES RELATIVE TO "UNIVERSAL REST FRAME"

Thus far it has been assumed tacitly that the center of mass of the solar system is at rest in the PPN coordinate system. Is this really a permissible assumption? Put differently, can one always so adjust the PPN coordinate system that its origin moves with any desired velocity (e.g., that of the solar system); or is the PPN coordinate system rigidly and irrevocably attached to some "universal rest frame"?
In general relativity, the geometry of curved spacetime picks out no preferred coordinate frames (except in cases with special symmetry). Therefore, one expects the velocity of the PPN coordinate frame to be freely specifiable. Put differently, one expects the entire PPN formalism, for general relativity, to be invariant under Lorentz transformations of the PPN coordinates [combined, perhaps, with "infinitesimal coordinate transformations" to maintain the gauge conditions that the "

Σ

" and "

U_{j k}

" terms of (39.24) and (39.23a) be absent]. By contrast, in Ni's theory of gravity (Box 39.1) the geometry of spacetime always picks out a preferred coordinate frame: the "rest frame of the universe." One would not be surprised, in this case, to find the PPN coordinate frame rigidly attached to the universal rest frame.
The above intuition is correct, according to calculations by Will (1971d) and by Will and Nordtvedt (1972). When dealing with general relativity and other theories with little or no "prior geometry," one can freely specify the velocity of the PPN coordinate system (at some initial instant of time). But for theories like Ni's, with a preferred "universal rest frame" ("preferred-frame theories"), only in the preferred frame can the post-Newtonian metric assume the form derived in the last section [equations (39.23)]. This restriction on the PPN metric does not mean that one is confined, in preferred-frame theories, to perform all calculations in the universal rest frame. Rather, it means that for such theories the PPN metric requires generalization to take account of coordinate-frame motion relative to the universal rest frame.
The required generalization can be achieved by subjecting the PPN metric (39.23) to (1) a Lorentz boost from the preferred frame

{x_{OLD}^{α}}

to a new PPN frame

{x_{NEW}^{α}}

, which moves with velocity

w

, plus (2) a change of gauge designed to keep the metric coefficients as simple as possible. The boost-plus-gauge-change is [Will and Nordtvedt (1972)]

Preferred-frame theories of gravity

Generalization of PPN metric to moving frames

\begin{aligned} (39.29a) & x_{OLD} = & x_{NEW} + \frac{1}{2} (x_{NEW} \cdot w) w + (1 + \frac{1}{2} w^{2}) w t_{NEW} \\ + O (ϵ^{5} t_{NEW} + ϵ^{4} x_{NEW}) \\ t_{OLD} = & t_{NEW} (1 + \frac{1}{2} w^{2} + \frac{3}{8} w^{4}) + (1 + \frac{1}{2} w^{2}) x_{NEW} \cdot w \\ (39.29b) & + \underset{⏟}{(\frac{1}{2} Δ_{2} + ζ - 1) w_{j} \frac{\partial χ}{\partial x_{NEW}^{j}}} + O (ϵ^{6} t_{NEW} + ϵ^{5} x_{NEW}) \\ (39.29c) & χ (t_{NEW}, x_{NEW}) \equiv & - \int ρ_{o} (t_{NEW}, x_{NEW}^{'}) | x_{NEW} - x_{NEW}^{'} | d^{3} x_{NEW}^{'} \end{aligned}

[Note: One insists, in the spirit of the post-Newtonian approximation, that the velocity

w

of the new PPN frame relative to the universal rest frame be no larger than the characteristic internal velocities of the system:

\begin{matrix} (39.30) & | w | ≲ ϵ .] \end{matrix}

This change of coordinates produces corresponding changes in the velocity of the matter

\begin{aligned} v_{OLD} = \frac{d x_{OLD}}{d t_{OLD}} = & v_{NEW} (1 - w \cdot v_{NEW} - \frac{1}{2} w^{2}) \\ (39.31) & + w (1 - \frac{1}{2} w \cdot v_{NEW}) + O (ϵ^{5}) \end{aligned}

A long but straightforward calculation (exercise 39.11) yields the following components for the metric in the new PPN coordinates. [Note: The subscripts NEW are here and hereafter dropped from the notation.]

\begin{matrix} (39.32a) & g_{j k} = δ_{j k} (1 + 2 γ U) + O (ϵ^{4}) \\ (39.32b) & g_{0 j} = - \frac{7}{2} Δ_{1} V_{j} - \frac{1}{2} Δ_{2} W_{j} + (α_{2} - \frac{1}{2} α_{1}) w_{j} U - α_{2} w_{k} U_{k j} + O (ϵ^{5}), \\ g_{00} = - 1 + 2 U - 2 β U^{2} + 4 Ψ - ζ Q - η D \\ (39.32c) & + (α_{2} + α_{3} - α_{1}) w^{2} U + (2 α_{3} - α_{1}) w_{j} V_{j} - α_{2} w_{j} w_{k} U_{j k} + O (ϵ^{6}) . \end{matrix}

Here

α_{1}, α_{2}

, and

α_{3}

are certain combinations of PPN parameters

\begin{matrix} (39.33a) & α_{1} = 7 Δ_{1} + Δ_{2} - 4 γ - 4 \\ (39.33b) & α_{2} = Δ_{2} + ζ - 1 \\ (39.33c) & α_{3} = 4 β_{1} - 2 γ - 2 - ζ \end{matrix}

The "gravitational potentials"

U, V_{j}, W_{j}, Ψ, a

, and

D

appearing here are to be calculated in the new, "moving" PPN coordinate system by the same prescriptions
as one used in the universal rest frame. Thus, their functional forms are the same as previously, but their values at any given event are different (see exercise 39.11):

\begin{matrix} (39.34a) & U (x, t) = \int \frac{ρ_{o} (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'}; \\ (39.34b) & V_{j} (x, t) = \int \frac{ρ_{o} (x^{'}, t) v_{j} (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'}; \\ (39.34c) & W_{j} (x, t) = \int \frac{ρ_{o} (x^{'}, t) [(x - x^{'}) \cdot v (x^{'}, t)] (x_{j} - x_{j}^{'}) d^{3} x^{'}}{{| x - x^{'} |}^{3}}; \\ Ψ (x, t) = \int \frac{ρ_{o} (x^{'}, t) ψ (x^{'}, t)}{| x - x^{'} |} d^{3} x^{'}, \\ (39.34d) & ψ = β_{1} v^{2} + β_{2} U + \frac{1}{2} β_{3} Π + \frac{3}{2} β_{4} p / ρ_{o}; \\ (39.34e) & a (x, t) = \int \frac{ρ_{o} (x^{'}, t) {[(x - x^{'}) \cdot v (x^{'}, t)]}^{2}}{{| x - x^{'} |}^{3}} d^{3} x^{'}; \\ (39.34f) & D (x, t) = \int \frac{[x - {x^{'} |}^{3}}{[t_{j \hat{k}} (x^{'}, t) - \frac{1}{3} δ_{j k} t_{l l} (x^{'}, t)] (x_{j} - x_{j}^{'}) (x_{k} - x_{k}^{'})} d^{3} x^{'} . \end{matrix}

The quantity

U_{j k}

is the gravitational potential defined in equation (39.23a):

\begin{matrix} (39.34~g) & U_{j k} (x, t) = \int \frac{ρ_{o} (x^{'}, t) (x_{j} - x_{j}^{'}) (x_{k} - x_{k}^{'})}{{| x - x^{'} |}^{3}} d^{3} x^{'} \end{matrix}

Notice that the velocity

w

of the PPN coordinate system relative to the universal rest frame appears explicitly in the PPN metric only if one or more of the coefficients

α_{1}, α_{2}, α_{3}

, is nonzero. Thus, theories with

α_{1} = α_{2} = α_{3} = 0

(e.g., general relativity) possess no preferred universal rest frame in the post-Newtonian limit; all their PPN frames are "created equal." By contrast, theories with at least one of

α_{1}, α_{2}, α_{3}

, nonzero (e.g., Ni's theory) do possess a preferred frame.

The generalized form (39.32) of the PPN metric, by virtue of the process used to construct it, is invariant under a Lorentz boost plus a gauge adjustment ["PostGalilean transformation"; see Chandrasekhar and Contopolous (1967)]:

\begin{aligned} x_{OLD} = x_{NEW} + \frac{1}{2} (x_{NEW} \cdot β) β + (1 + \frac{1}{2} β^{2}) β t_{NEW} \\ (39.35) & + O (ϵ^{5} t_{NEW} + ϵ^{4} x_{NEW}) \\ t_{OLD} = (1 + \\ + \frac{1}{2} β^{2} + \frac{3}{8} β^{4}) t_{NEW} + (1 + \frac{1}{2} β^{2}) x_{NEW} \cdot β \\ + (\frac{1}{2} Δ_{2} + ζ_{1} - 1) β \cdot \nabla_{NEW} χ + O (ϵ^{6} t_{NEW} + ϵ^{5} x_{NEW}) \end{aligned}

Of course, it is also invariant under spatial rotations.

EXERCISE

Exercise 39.11. TRANSFORMATION TO MOVING FRAME

Show that the change of coordinates (39.29) changes the PPN metric coefficients from the form (39.23) to the form (39.32). [Hints: (1) Keep firmly in mind the fact that the potentials

U, V_{j}, W_{j}, A

, and

D

are not scalar fields. Each coordinate system possesses its own potentials. For example, by using equations (39.29) in the integral for

U_{oLD}

, one finds

\begin{aligned} (39.36) & U_{OLD} (x_{OLD}, t_{OLD}) & = \int \frac{ρ_{o} (x_{OLD}^{'}, t_{OLD})}{| x_{OLD} - x_{OLD}^{'} |} d^{3} x_{OLD}^{'} \\ = {[U_{NEW} - w_{j} (V_{j NEW} - W_{j NEW}) + \frac{1}{2} w_{j} w_{k} x_{, j k}]}_{x_{NEW}, t_{NBW}} + O (ϵ^{6}) . \end{aligned}

(2) The law of baryon conservation (39.44) may be useful.]

§39.10. PPN STRESS-ENERGY TENSOR

The motion of the solar system is governed by the equations

T^{α β}; β = 0

. Before studying them, one must calculate the post-Newtonian corrections to the stress-energy tensor in the PPN coordinate frame. This requires a transformation from the comoving, orthonormal frame

ω^{\hat{α}}

, where

\begin{matrix} (39.37) & T^{\hat{0} \hat{0}} = ρ_{o} (1 + Π), T^{\hat{0} \hat{j}} = 0, T^{\hat{j} \hat{k}} = t_{j \hat{k}} \end{matrix}

to the coordinate frame. One can effect this transformation in two stages: stage 2 is a transformation

\begin{matrix} (39.38a) & ω^{\tilde{0}} \equiv [1 - U + O (ϵ^{4})] d t \\ + [\frac{7}{2} Δ_{1} V_{j} + \frac{1}{2} Δ_{2} W_{j} + (\frac{1}{2} α_{1} - α_{2}) w_{j} U + α_{2} w_{k} U_{k j} + O (ϵ^{5})] d x^{j} \\ (39.38b) & ω^{\tilde{j}} \equiv [(1 + γ U) δ_{j k} + O (ϵ^{4})] d x^{k} + O (ϵ^{5}) d t \end{matrix}

between the coordinate frame and an orthonormal frame attached to it; stage 1 is a pure Lorentz transformation (boost) between the two orthonormal frames

ω^{\tilde{α}}

and

ω^{\hat{α}}

. The 4 -velocity of this boost is minus the 4 -velocity of the matter, which has components

\begin{matrix} (39.40) & \begin{matrix} u^{j} = v_{j} u^{0}, u^{0} = 1 + \frac{1}{2} v^{2} + U + O (ϵ^{4}) in coord. frame; \\ u^{\tilde{j}} = v_{\tilde{j}} u^{\tilde{o}}, u^{\tilde{0}} = 1 + \frac{1}{2} v^{2} + O (ϵ^{4}), \\ v_{\tilde{j}} = v_{j} [1 + (1 + γ) U] \end{matrix}} in ω^{\tilde{α}} frame. \end{matrix}

Combining the boost, which has ordinary velocity

β_{\tilde{j}} = - v_{\tilde{j}}

, with the transformation (39.38), and then inverting, one obtains the result (exercise 39.12)

\begin{aligned} d x^{α} = A^{α}_{β} ω^{\hat{β}}, {\begin{matrix} ω^{\hat{β}} = orthonormal comoving basis, \\ d x^{α} = PPN coordinate basis; \end{matrix} \\ A^{0}_{\hat{o}} = 1 + \frac{1}{2} v^{2} + U + O (ϵ^{4}), \\ A^{0}_{j} = v_{j} [1 + \frac{1}{2} v^{2} + (2 + γ) U] - \frac{7}{2} Δ_{1} V_{j} - \frac{1}{2} Δ_{2} W_{j} \\ (39.41) & + (α_{2} - \frac{1}{2} α_{1}) w_{j} U - α_{2} w_{k} U_{k j} + O (ϵ^{5}), \\ A^{j}_{\hat{0}} = v_{j} [1 + \frac{1}{2} v^{2} + U] + O (ϵ^{5}), \\ A^{j}_{\hat{k}} = (1 - γ U) δ_{j k} + \frac{1}{2} v_{j} v_{k} + O (ϵ^{4}) . \end{aligned}

Transformation from rest frame of matter to PPN coordinate frame

This transformation, when applied to the stress-energy tensor (39.37) yields, in the PPN coordinate frame,

\begin{aligned} (39.42a) & T^{00} = & ρ_{o} (1 + Π + v^{2} + 2 U) + O (ρ_{o} ϵ^{4}) \\ (39.42b) & T^{0 j} = & ρ_{o} (1 + Π + v^{2} + 2 U) v_{j} + t_{\hat{j} \hat{m}} v_{m} + O (ρ_{o} ϵ^{5}) \\ T^{j k} = & t_{j \hat{k}} (1 - 2 γ U) + ρ_{o} (1 + Π + v^{2} + 2 U) v_{j} v_{k} \\ (39.42c) & + \frac{1}{2} (v_{j} t_{\hat{k} \hat{m}} v_{m} + v_{k} t_{\hat{j} \hat{m}} v_{m}) + O (ρ_{o} ϵ^{6}) \end{aligned}

Exercise 39.12. THE TRANSFORMATION BETWEEN COMOVING FRAME AND PPN FRAME

Carry out the details of the derivation of the transformation matrix (39.41); and in the process calculate the correction of

O (ϵ^{4})

A^{0}_{0}

§39.11. PPN EQUATIONS OF MOTION

The post-Newtonian corrections to the Newtonian equations of motion (39.15) and (39.16) are derived from the law of conservation of baryon number

{(ρ_{o} u^{α})}_{; α} = 0

, and from the law of conservation of local energy-momentum,

T^{α β}_{; β} = 0

. The simplest of the equations of motion is the conservation of baryon number. Its exact expression is

{(ρ_{o} u^{α})}_{; α} = (1 / \sqrt{- g}) {(\sqrt{- g} ρ_{o} u^{α})}_{, α} = 0

. Define a new quantity

\begin{aligned} (39.43) & ρ^{*} & \equiv ρ_{o} (1 + \frac{1}{2} v^{2} + 3 γ U) \\ = ρ_{o} u^{0} \sqrt{- g} + O (ρ_{o} ϵ^{4}) \end{aligned}

Law of energy conservation

Post-Newtonian Euler equation
[see (39.39) for

u^{0}

, and (39.32) for the metric]. Then rest-mass conservation takes on the same form as at the Newtonian order (39.15a), except now it is more accurate:

\begin{matrix} (39.44) & ρ_{, t}^{*} + {(ρ^{*} v_{j})}_{, j} = 0 + errors of O (ρ_{o, j} ϵ^{5}) \end{matrix}

The next simplest equation of motion is

T_{; α}^{0 α} = 0

. Straightforward evaluation, using the metric of equations (39.32) and the stress-energy tensor of equations (39.42), yields

\begin{aligned} {[ρ_{o} (1 + Π + v^{2} + 2 U)]}_{, t} & + {[ρ_{o} (1 + Π + v^{2} + 2 U) v_{j} + t_{\hat{j} \hat{m}} v_{m}]}_{, j} \\ (39.45) & + (3 γ - 2) ρ_{o} U_{, t} + (3 γ - 3) ρ_{o} v_{k} U_{, k} = O (ρ_{o, j} ϵ^{5}) . \end{aligned}

By subtracting equation (39.44) from this, and using the Newtonian equations of motion (39.15) and (39.16) to simplify several terms where the Newtonian approximation is adequate, one obtains

\begin{matrix} (39.46) & ρ_{o} d Π / d t + t_{j \hat{k}} v_{j, k} = 0 + errors of O (ρ_{o, j} ϵ^{5}) \end{matrix}

Notice that this is nothing but the first law of thermodynamics (local energy conservation) with energy flow through the matter being neglected. (Neglecting energy flow was justified in §39.5.) This first law of thermodynamics is actually a post-Newtonian equation in the context of hydrodynamics, rather than a Newtonian equation, because

Π

does not affect the hydrodynamic motion at Newtonian order (see §39.7).

The last of the equations of motion,

T^{j α}_{; α} = 0

, reduces to the post-Newtonian Euler equation

\begin{aligned} ρ^{*} \frac{d v_{j}}{d t} & - ρ^{*} U_{, j} + {[t_{\hat{j} \hat{k}} (1 + 3 γ U)]}_{, k} - t_{\hat{j} \hat{k}, k} (\frac{1}{2} v^{2} + Π) - \frac{t_{\hat{j} k} t_{\hat{k} ℓ, ℓ}}{ρ^{*}} \\ + ρ^{*} \frac{d}{d t} [(2 γ + 2) U v_{j} - \frac{1}{2} (7 Δ_{1} + Δ_{2}) V_{j} - \frac{1}{2} α_{1} U w_{j}] - v_{j} ρ^{*} U_{, t} + v_{k} t_{\hat{k} \hat{j}, t} \\ + \frac{1}{2} Δ_{2} ρ^{*} {(V_{j} - W_{j})}_{, t} + \frac{1}{2} ρ^{*} [(7 Δ_{1} + Δ_{2}) v_{k} + (α_{1} - 2 α_{3}) w_{k}] V_{k, j} \\ (39.47) & - ρ^{*} {[2 Ψ - \frac{1}{2} ζ a - \frac{1}{2} η D - \frac{1}{2} α_{2} w_{i} w_{k} U_{i k} + α_{2} w_{i} (V_{i} - W_{i})]}_{, j} \\ - ρ^{*} U_{, j} [γ v^{2} - \frac{1}{2} α_{1} w \cdot v + \frac{1}{2} (α_{2} + α_{3} - α_{1}) w^{2} - (2 β - 2) U + 3 γ p / ρ^{*}] \\ + \frac{1}{2} (v_{j, k} t_{\hat{k} \hat{m}} v_{m} - t_{\hat{j} \hat{m}} v_{m, k} v_{k}) + \frac{1}{2} [v_{m} {(t_{\hat{m} \hat{j}} v_{k})}_{, k} - v_{j} {(t_{\hat{k} \hat{ı}} v_{k})}_{, ℓ}] = 0 . \end{aligned}

Partial derivatives are denoted by commas;

d / d t

is the time-derivative following the matter [equation (39.16)].

Equations (39.44), (39.46), and (39.47) are a complete set of equations of motion at the post-Newtonian order.

Exercise 39.13. EQUATIONS OF MOTION

EXERCISES
Carry out the details of the derivation of the equations of motion (39.44), (39.46), and (39.47). As part of the derivation, calculate the following values of the Christoffel symbols in the PPN coordinate frame:

\begin{aligned} Γ^{0}_{00} = & - U_{, t} + O (U_{, j} ϵ^{3}), Γ_{0 j}^{0} = - U_{, j} + O (U_{, j} ϵ^{2}), \\ Γ^{0}_{j k} = & γ U_{, t} δ_{j k} + \frac{7}{2} Δ_{1} V_{(j, k)} + \frac{1}{2} Δ_{2} W_{(j, k)} + (\frac{1}{2} α_{1} - α_{2}) w_{(j} U_{, k)} \\ + α_{2} w_{i} U_{i (j, k)} + O (U_{, j} ϵ^{3}) . \\ Γ^{j}_{00} = & - U_{, j} + [(β + γ) U^{2} - 2 Ψ + \frac{1}{2} ζ a + \frac{1}{2} η η^{D} + \frac{1}{2} (α_{1} - α_{2} - α_{3}) w^{2} U \\ (39.48) & {+ \frac{1}{2} (α_{1} - 2 α_{3}) w_{i} V_{i} + \frac{1}{2} α_{2} w_{i} w_{k} U_{i k}]}_{, j} - \frac{7}{2} Δ_{1} V_{j, t} - \frac{1}{2} Δ_{2} W_{j, t} \\ + (α_{2} - \frac{1}{2} α_{1}) w_{j} U_{, t} - α_{2} w_{i} U_{i j, t} + O (U_{, j} ϵ^{4}), \\ Γ^{j}_{0 k} = & γ U_{, t} δ_{j k} - (\frac{7}{2} Δ_{1} + \frac{1}{2} Δ_{2}) V_{[j, k]} - \frac{1}{2} α_{1} w_{[j} U_{, k]} + O (U_{, j} ϵ^{3}), \\ Γ^{j}_{k ℓ} = & - γ (U_{, j} δ_{k ℓ} - 2 U_{, (k} δ_{ℓ j}) + O (U_{, j} ϵ^{2}) . \end{aligned}

Here square brackets on tensor indices denote antisymmetrization, and round brackets denote symmetrization. As part of the derivation, it may be useful to prove and use the relations

\begin{aligned} (39.49a) & χ (t, x) & = - \int ρ_{o} (t, x^{'}) | x - x^{'} | d^{3} x^{'} \\ (39.49b) & χ_{, j k} & = - δ_{j k} U + U_{j k} \\ (39.49c) & χ_{, i t} & = V_{i} - W_{i} + O (ϵ^{5}) \\ (39.50) & W_{[k, j]} & = V_{[k, j]} . \end{aligned}

Here

χ

is the function originally defined in equation (39.29c).
Exercise 39.14. POST-NEWTONIAN APPROXIMATION TO GENERAL RELATIVITY
Perform a post-Newtonian expansion of Einstein's field equations, thereby obtaining the values cited in Box 39.2 for the PPN parameters of general relativity. The calculations might best follow the approach of Chandrasekhar (1965a): Set

g_{α β} = η_{α β} + h_{α β}

, and assume

\begin{matrix} (39.51) & h_{00} = O (ϵ^{2}) + O (ϵ^{4}), h_{0 j} = O (ϵ^{3}), h_{j k} = O (ϵ^{2}) \end{matrix}

Choose the space and time coordinates so that the four "gauge conditions"

\begin{matrix} (39.52) & \begin{array}{l} h_{j k, k} - \frac{1}{2} h_{, j} = O (ϵ^{4} / R_{⊙}) \\ h_{0 k, k} - \frac{1}{2} h_{k k, 0} = O (ϵ^{5} / R_{⊙}) \end{array}} with h = h_{α β} η^{α β} = - h_{00} + h_{l ℓ} \end{matrix}

are satisfied.
(a) Show that the spatial gauge conditions are the post-Newtonian approximations to those (35.1a) used in the study of weak gravitational waves, but that the temporal gauge condition is not.
(b) Use these gauge conditions and the post-Newtonian limit in equations (8.24) and (8.47) to obtain for the Ricci tensor, accurate to linearized order,

\begin{matrix} (39.53a) & R_{00} = - \frac{1}{2} h_{00, m m} + O (ϵ^{4} / R_{⊙}^{2}), R_{j k} = - \frac{1}{2} h_{j k, m m} + O (ϵ^{4} / R_{⊙}^{2}) \\ (39.53b) & R_{0 j} = - \frac{1}{2} h_{0 j, m m} - \frac{1}{4} h_{00, 0 j} + O (ϵ^{5} / R_{⊙}^{2}) \end{matrix}

(c) Combine these with the Newtonian form (39.13) of the stress-energy tensor, and with equation (39.27), to obtain the following metric coefficients, accurate to linearized order:

\begin{matrix} (39.54) & h_{00} = 2 U + k_{00} + O (ϵ^{6}), h_{0 j} = - \frac{7}{2} V_{j} - \frac{1}{2} W_{j} + O (ϵ^{5}), \\ [unknown post-Newtonian correction] \\ h_{j k} = 2 U δ_{j k} + O (ϵ^{4}) . \end{matrix}

Here

U, V_{j}

, and

W_{j}

are to be regarded as defined by equations (39.34a,b,c). By comparing these metric coefficients with equations (39.32), discover that

\begin{matrix} (39.55) & γ = 1, Δ_{1} = 1, Δ_{2} = 1 \end{matrix}

for general relativity.
(d) With this knowledge of the metric in linearized order, one can carry out the analysis of

§ 39.10

(using

γ = Δ_{1} = Δ_{2} = 1

throughout), to obtain the post-Newtonian corrections to the stress-energy tensor [equation (39.42) with

γ = 1

].
(e) Calculate, similarly, the post-Newtonian corrections to the Ricci tensor component

R_{00}

, using

g_{α β} = η_{α β} + h_{α β}

, using

h_{α β}

as given in equations (39.54), and using the gauge conditions (39.52). The answer should be

\begin{matrix} (39.56) & R_{00} = {(- U - \frac{1}{2} k_{00} - U^{2})}_{, m m} + 4 U U_{, m m} + O (ϵ^{6} / R_{⊙}^{2}) \end{matrix}

(f) Evaluate the Einstein equation

R_{00} = 8 π (T_{00} - \frac{1}{2} g_{00} T)

, accurate to post-Newtonian order, and solve it to obtain the post-Newtonian metric correction

\begin{matrix} (39.57) & k_{00} = - 2 U^{2} + 4 Ψ \end{matrix}

where

Ψ

is given by equations (39.34d) with

β_{1} = β_{2} = β_{3} = β_{4} = 1

. By comparing with equations (39.32c) and (39.34d), discover that

\begin{matrix} (39.58) & β = β_{1} = β_{2} = β_{3} = β_{4} = 1, ζ = η = 0 \end{matrix}

for general relativity.
(g) Knowing the full post-Newtonian metric, and the full post-Newtonian stress-energy tensor, one can carry out the calculations of

§ 39.11

(using

γ = β = β_{1} = β_{2} = β_{3} = β_{4} =

Δ_{1} = Δ_{2} = 1, ζ = η = 0

) to obtain the post-Newtonian equations of motion for the matter [equations (39.44), (39.46), and (39.47)].

§39.12. RELATION OF PPN COORDINATES TO SURROUNDING UNIVERSE

One crucial issue remains to be clarified: What is the orientation of the PPN coordinate system relative to the surrounding universe? More particularly: Does the PPN coordinate system rotate relative to the "fixed stars on the sky;" or is it "rigidly attached" to them, in some sense? In order to answer this question, imagine using the PPN formalism to analyze the solar system. Make no assumptions about the solar system's velocity through the PPN coordinate frame. Then, as one moves outward from the Sun, past the Earth's orbit, past Pluto's orbit, and on out toward interstellar space, one sees the PPN coordinate frame become more and more Lorentz in its global properties

[g_{α β} = η_{α β} + O (M_{⊙} / r)

]. Thus, far from the solar system the PPN coordinates become a "Lorentz frame moving through the galaxy." This means, of course, that the spatial axes of the PPN coordinate frame behave as though they were attached to gyroscopes far outside the solar system. Equivalently: The PPN coordinate system Fermi-Walker-transports its spatial axes through the spacetime geometry of the galaxy and universe.

§39.13. SUMMARY OF PPN FORMALISM

The PPN formalism, as constructed in this chapter, is summarized in Box 39.4. Much of the recent literature uses a different set of PPN parameters than are used in this book; for a translation from one parameter set to the other, see Box 39.5.

Exercise 39.15. MANY-BODY SYSTEM IN POST-NEWTONIAN LIMIT OF GENERAL RELATIVITY

Consider, in the post-Newtonian limit of general relativity, a system made up of many gravitationally interacting bodies with separations large compared to their sizes (example: the solar system). Idealize each body to be spherically symmetric, to be free of internal motions, and to have isotropic internal stresses,

t_{j k} = δ_{j k} p

. Let the world line of the center of body

A

, in some chosen PPN coordinate frame, be

x_{A} (t)

; and let the (coordinate) velocity of the center of body

A

\begin{matrix} (39.59a) & v_{A} (t) = d x_{A} / d t . \end{matrix}

The total mass-energy of body

A

as measured in its neighborhood (rest mass-energy plus internal energy plus self-gravitational energy) is given by

\begin{matrix} (39.59b) & M_{A} = \int_{V_{A}} (1 + Π - \frac{1}{2} U_{self}) d (rest mass) + O (M_{A} ϵ^{4}) \end{matrix}

where

U_{self}

is the body's own Newtonian potential (no contributions from other bodies), and

V_{A}

is the interior of the body.

Solar system's PPN coordinate frame is attached to a local Lorentz frame of Galaxy

Box 39.4 SUMMARY OF THE PPN FORMALISM

I. Variables

ρ_{o} (x, t)

: baryon "mass" density (

§ 39.3)

, as measured in rest frame

Π (x, t)

: specific internal energy (dimensionless;

§ 39.3

), as measured in rest frame

t_{j \hat{k}} (x, t)

: components of stress referred to orthonormal axes of rest frame

v_{j} (x, t)

: coordinate velocity of matter (i.e., rest frame) relative to PPN coordinates

U (x, t), Ψ (x, t), A (x, t), D (x, t), V_{j} (x, t), W_{j} (x, t), U_{j k} (x, t)

: gravitational potentials

γ, β, β_{1}, β_{2}, β_{3}, β_{4}, Δ_{1}, Δ_{2}, ζ, η

: parameters whose values distinguish one theory from another (see Box 39.2)

w

: velocity of PPN coordinate frame relative to "universal rest frame" [relevant only for theories with nonzero

α_{1}, α_{2}

, or

α_{3}

; see eq. (39.33)].
II. Equations governing evolution of these variables

ρ_{o}

: conservation of rest mass, equation (39.44)

Π

: first law of thermodynamics, equation (39.46)

t_{j k}

: determined in terms of

ρ_{o}, Π

, and other material variables (chemical composition, strains, etc.) by equations of state and the usual theory of a stressed medium-which is not discussed here

v_{j}

: equations of motion ("

F = m a

"), equations (39.47)

U, Ψ, A, D, V_{j}, W_{j}, U_{j k}

: source equations (39.34)
III. Quantities to be calculated from these variables

g_{00} (x, t), g_{0 j} (x, t), g_{j k} (x, t)

: these components of metric in PPN coordinate frame are expressed in terms of gravitational potentials by equations (39.32)

u^{0} (x, t), u^{j} (x, t)

: these components of matter 4 -velocity in PPN coordinate frame are given by equations (39.39)

T^{00} (x, t), T^{0 j} (x, t), T^{j k} (x, t)

: these components of stress-energy tensor in PPN coordinate frame are given by equations (39.42)
IV. Relation between rest frame, PPN coordinates, and the universe

Orthonormal basis $ω^{\hat{α}}$ of rest frame, where $t_{j \hat{k}}$ are defined, is related to PPN coordinate basis $d x^{α}$ by equations (39.41)
Far from the sun, the PPN coordinates become asymptotically Lorentz; i.e., they form an inertial frame moving through the spacetime geometry of the galaxy and the universe.
Gives no account of expansion of universe or of cosmic gravitational waves impinging on solar system.

Box 39.5 PPN PARAMETERS USED IN LITERATURE: A TRANSLATOR'S GUIDE

The original "point-particle version" of the PPN formalism [Nordtvedt (1968b)], and the original "perfect-fluid version" [Will (1971c)] used different sets of PPN parameters. This book has adopted Will's set, and has added the parameter

η

characterizing effects of anisotropic stresses. More recently, Will and Nordtvedt have jointly adopted a revised set of parameters, described below.
A. Translation Table

Will-Nordtvedt

revised parameters

Revised parameters in

notation of this book

Revised parameters in

notation of Nordtvedt

(1968b)^{c}

γ

γ

γ

β

β

β

α_{1}

7 Δ_{1} + Δ_{2} - 4 γ - 4

8 Δ - 4 γ - 4

α_{2}

Δ_{2} + ζ - 1

α^{'''} - 1

α_{3}

4 β_{1} - 2 γ - 2 - ζ

4 α^{''} - α^{'''} - 2 γ - 1

ζ_{1}

ζ

α^{'''} - χ

ζ_{2}

2 β + 2 β_{2} - 3 γ - 1

2 β - α^{'} - 1

ζ_{3}

β_{3} - 1

absent

ζ_{4}

β_{4} - γ

absent

^{a}

Revised parameters are used by Will and Nordtvedt (1972), Nordtvedt and Will (1972), Will (1972), and Ni (1973).

^{b}

Notation of this book is used by Will (1971a,b,c,d), Ni (1972), and Thorne, Ni, and Will (1971).

^{c}

Nordtvedt's original "point-particle" parameters were used by Nordtvedt (1968b, 1970, 1971a,b).

B. Significance of Revised Parameters

α_{1}, α_{2}, α_{3}

measure the extent of and nature of "preferred-frame effects"; see

§ 39.9

. Any theory of gravity with at least one

α

nonzero is called a preferred-frame theory.

ζ_{1}, ζ_{2}, ζ_{3}, ζ_{4}, α_{3}

measure the extent of and nature of breakdowns in global conservation laws. A theory of gravity possesses, at the post-Newtonian level, all 10 global conservation laws ( 4 for energy-momentum, 6 for angular momentum; see Chapters 19 and 20) if and only if

ζ_{1} = ζ_{2} = ζ_{3} = ζ_{4} = α_{3} = 0

. See Will (1971d), Will and Nordtvedt (1972), Will (1972), for proofs and discussion. Any theory with

ζ_{1} = ζ_{2}

= ζ_{3} = ζ_{4} = α_{3} = 0

is called a conservative theory.
In general relativity and the Dicke-Brans-Jordan theory, all

α

's and

ζ

's vanish. Thus, general relativity and Dicke-Brans-Jordan are conservative theories with no preferred-frame effects.
(a) Show that, when written in the chosen PPN coordinate frame, this expression for

M_{A}

becomes

\begin{matrix} (39.59c) & M_{A} = \int_{V_{A}} ρ_{o} (1 + Π + \frac{1}{2} v_{A}^{2} + 3 U - \frac{1}{2} U_{self}) d^{3} x + O (M_{A} ϵ^{4}) \end{matrix}

Use equations (39.43), (39.44), and (39.46) to show that

M_{A}

is conserved as the bodies move about,

d M_{A} / d t = 0

.
(b) Pick an event

(t, x)

outside all the bodies, and at time

t

denote

\begin{matrix} (39.59d) & r_{A} \equiv x_{A} - x, r_{A B} \equiv x_{A} - x_{B}, r_{A} \equiv | r_{A} |, r_{A B} \equiv | r_{A B} | \end{matrix}

Show that the general-relativistic, post-Newtonian metric (39.32) at the chosen event has the form

\begin{aligned} (39.60a) & g_{i k} = δ_{j k} (1 + 2 \sum_{A} \frac{M_{A}}{r_{A}}) + O (ϵ^{4}), \\ (39.60b) & g_{0 j} = - \sum_{A} \frac{M_{A}}{r_{A}} [\frac{7}{2} v_{A j} + \frac{1}{2} \frac{(v_{A} \cdot r_{A}) r_{A j}}{r_{A}^{2}}] + O (ϵ^{5}), \\ g_{00} = \\ (39.60c) & - 1 + 2 \sum_{A} \frac{M_{A}}{r_{A}} - 2 {(\sum_{A} \frac{M_{A}}{r_{A}})}^{2} + 3 \sum_{A} \frac{M_{A} v_{A}^{2}}{r_{A}} \\ - 2 \sum_{A} \sum_{B \neq A} \frac{M_{A} M_{B}}{r_{A} r_{A B}} + O (ϵ^{6}) . \end{aligned}

[Hint: From the Newtonian virial theorem (39.21a), applied to body

A

by itself in its own rest frame, conclude that

\begin{matrix} (39.61) & \int_{{\tilde{T}}_{A}} (3 p - \frac{1}{2} ρ_{o} U_{self}) d^{3} x = O (M_{A} ϵ^{4}), \end{matrix}

where the integral is performed in the PPN frame.]
(c) Perform an infinitesimal coordinate transformation,

\begin{matrix} (39.62) & t_{OLD} = t_{NEW} - \frac{1}{2} \sum_{A} \frac{M_{A} (r_{A} \cdot v_{A})}{r_{A}}, x_{OLD} = x_{NEW} \end{matrix}

to bring the metric (39.60) into the standard form originally devised by Einstein, Infeld, and Hoffman (1938), and by Eddington and Clark (1938):

\begin{aligned} (39.63a) & g_{j k} = δ_{j k} (1 + 2 \sum_{A} \frac{M_{A}}{r_{A}}) + O (ϵ^{4}), \\ (39.63b) & g_{0 j} = - 4 \sum_{A} \frac{M_{A}}{r_{A}} v_{A j} + O (ϵ^{5}), \\ g_{00} = - 1 + 2 \sum_{A} \frac{M_{A}}{r_{A}} - 2 {(\sum_{A} \frac{M_{A}}{r_{A}})}^{2} + 3 \sum_{A} \frac{M_{A} v_{A}^{2}}{r_{A}} \\ (39.63c) & - 2 \sum_{A} \sum_{B \neq A} \frac{M_{A} M_{B}}{r_{A} r_{A B}} - \frac{\partial^{2} χ}{\partial t^{2}} + O (ϵ^{6}), \end{aligned}

where

χ

[equation (39.49a)] is given by

χ = - \sum_{A} M_{A} r_{A}

(d) The equations of motion for the bodies can be obtained in either of two ways: by performing a volume integral of the Euler equation (39.48) over the interior of each body; or by invoking the general arguments of

§ 20.6

. The latter way is the easier. Use it to conclude that any chosen body

K

moves along a geodesic of the metric obtained by omitting the terms

A = K

from the sums in (39.63). Show that the geodesic equation for body

K

reduces to

\begin{aligned} \frac{d^{2} x_{K}}{d t^{2}} \equiv \frac{d v_{K}}{d t} = & \sum_{A \neq K} r_{A K} \frac{M_{A}}{r_{A K}^{3}} [1 - 4 \sum_{B \neq K} \frac{M_{B}}{r_{B K}} - \sum_{C \neq A} \frac{M_{C}}{r_{C A}} (1 - \frac{r_{A K} \cdot r_{C A}}{2 r_{C A}^{2}}) \\ + v_{K}^{2} + 2 v_{A}^{2} - 4 v_{A} \cdot v_{K} - \frac{3}{2} {(\frac{v_{A} \cdot r_{A K}}{r_{A K}})}^{2}] \\ (39.64) & - & \sum_{A \neq K} (v_{A} - v_{K}) \frac{M_{A} r_{A K} \cdot (3 v_{A} - 4 v_{K})}{r_{A K}^{3}} \\ + & \frac{7}{2} \sum_{A \neq K} \sum_{C \neq A} r_{C A} \frac{M_{A} M_{C}}{r_{A K} r_{C A}^{3}} \end{aligned}

Equations (39.63) and (39.64) are called the Einstein-Infeld-Hoffman ("EIH") equations for the geometry and evolution of a many-body system. They are used widely in analyses of planetary orbits in the solar system. For example, the Caltech Jet Propulsion Laboratory uses them, in modified form, to calculate ephemerides for high-precision tracking of planets and spacecraft. The above method of deriving the EIH equations and metric was devised by Fock (1959). For a similar calculation in the Dicke-Brans-Jordan theory, see Estabrook (1969); and for a derivation of the analogous many-body equations in the full PPN formalism, see Will (1972).

сниетеп 40

SOLAR-SYSTEM EXPERIMENTS

This chapter analyzes experiments using PPN formalism

Complexity of solar system's spacetime geometry

§40.1. MANY EXPERIMENTS OPEN TO DISTINGUISH GENERAL RELATIVITY FROM PROPOSED METRIC THEORIES OF GRAVITY

No audience will show up for a fight if in everyone's eyes the challenger has zero chance to win. No battle-hungry promoter desperately trying to finance the fight can afford to put into the ring against the champion any but the best contender that he can find. Against Einstein's metric theory of gravity, the judgment of the day (as

$ 39.2

showed) leaves one no option except to put up another theory of gravity that is also metric (or metric plus torsion).

To put on a contest, then, is to design and perform an experiment that distinguishes general relativity from some not completely implausible metric theory of gravity. This chapter describes such experiments-some already performed; some to be performed in the future-and analyses their significance using the PPN formalism of Chapter 39.

In most of the experiments to be described, one investigates the motion of the moon, planets, spacecraft, light rays, or gyroscopes through the spacetime geometry of the solar system. That spacetime geometry is very complicated. It includes the spherical fields of the sun and all the planets, nonspherical fields due to their quadrupolar and higher-order deformations, and fields due to their momentum and angular momentum. Moreover, the spacetime geometry results-or at least in the post-Newtonian formalism it is viewed as resulting-from a nonlinear superposition of all these fields.*

Fortunately for this discussion, several of the most important experiments are free of almost all these complications. The effects they measure are associated entirely with the spherical part of the sun's gravitational field. A description of these experiments will come first (

$ § 40.2 - 40.5

), and then attention will turn to experiments that are more complex in principle.

To discuss central-field experiments, one needs an expression for the external gravitational field of an idealized, isolated, static, spherical sun. In general relativity, such a gravitational field is described by the Schwarzschild line element,

d s^{2} = - (1 - \frac{2 M_{⊙}}{r}) d t^{2} + \frac{d r^{2}}{1 - 2 M_{⊙} / r} + r^{2} (d θ^{2} + \sin^{2} ϕ d ϕ^{2})

But this line element is not what one wants, for two reasons: (1) it is "too accurate"; (2) it is written in the "wrong" coordinate system.

Why too accurate? Because it is simple only when unperturbed and unmodified; whereas some modified theories show up new effects that are so complex they are tractable only in the post-Newtonian approximation. Why wrong coordinate system? Because physicists, astronomers, and other celestial mechanics have adopted the fairly standard convention of using "isotropic coordinates" rather than "Schwarzschild coordinates" when analyzing the solar system. Example: post-Newtonian expansions, including the PPN formalism of Chapter 39, almost always use isotropic coordinates. Another example: the relativistic ephemeris for the solar system, prepared by the Caltech Jet Propulsion Laboratory [Ohandley et al. (1969); Anderson (1973)] and used extensively throughout the world, employs isotropic coordinates.

Modify the Schwarzschild line element, then. First transform to isotropic coordinates (Exercise 31.7); then expand the metric coefficients in powers of

M_{⊙} / r

, to post-Newtonian accuracy. Thereby obtain

\begin{aligned} d s^{2} & = - [1 - 2 \frac{M_{⊙}}{r} + 2 {(\frac{M_{⊙}}{r})}^{2}] d t^{2} + [1 + 2 \frac{M_{⊙}}{r}] [d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})] \\ (40.1) & = - [1 - 2 \frac{M_{⊙}}{r} + 2 {(\frac{M_{⊙}}{r})}^{2}] d t^{2} + [1 + 2 \frac{M_{⊙}}{r}] [d x^{2} + d y^{2} + d z^{2}] . \end{aligned}

Here

r, θ, ϕ

are related to

x, y, z

in the usual manner:

\begin{matrix} (40.2) & r = {(x^{2} + y^{2} + z^{2})}^{1 / 2}, θ = \tan^{- 1} [z / {(x^{2} + y^{2})}^{1 / 2}], ϕ = \tan^{- 1} (y / x); \end{matrix}

and

r

is the new, "isotropic" radial coordinate, not to be confused with the Schwarzschild

r

. (The reader who has not studied

$ 39.6

will discover in the next section why one keeps terms of order

M_{⊙}^{2} / r^{2}

g_{00}

but not in

g_{j k}

.) Note: this postNewtonian expression for the metric is a special case of the result derived in exercise 19.3.

If one calculates the gravitational field of the same source (the sun) in the same post-Newtonian approximation in other metric theories of gravity, one obtains a very similar result:

Idealization of geometry to that of isolated, static, spherical sun:
(1) in Schwarzschild coordinates
(2) in isotropic coordinates
(3) in PPN formalism
(4) including preferred-frame effects

\begin{aligned} d s^{2} & = - [1 - 2 \frac{M_{⊙}}{r} + 2 β {(\frac{M_{⊙}}{r})}^{2}] d t^{2} + [1 + 2 γ \frac{M_{⊙}}{r}] [d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})] \\ (40.3) & = - [1 - 2 \frac{M_{⊙}}{r} + 2 β {(\frac{M_{⊙}}{r})}^{2}] d t^{2} + [1 + 2 γ \frac{M_{⊙}}{r}] [d x^{2} + d y^{2} + d z^{2}] \end{aligned}

(see exercise 40.1). Here

γ

and

β

are two of the ten PPN parameters described in Box 39.2. Recall from that box that

γ

measures "the amount of space curvature produced by unit rest mass," while

β

measures "the amount of nonlinearity in the superposition law for

g_{00}

." These heuristic descriptions find their mathematical counterparts in the above form for the idealized metric surrounding a spherically symmetric center of attraction.

By measuring the parameter

γ

to high precision, one can distinguish between general relativity

(γ = 1)

and the Dicke-Brans-Jordan theory

[γ = (1 + ω) / (2 + ω)

, where

ω

is the "Dicke coupling constant"]; see Box 39.2. But general relativity and Dicke-Brans-Jordan predict the same value for

β (β = 1)

. This identity does not mean that

β

is unworthy of measurement. A value

β \neq 1

is predicted by other theories [see Ni (1972)]; so measurements of

β

are useful in distinguishing such theories from general relativity.

Actually, the above form (40.3) for the sun's metric is not fully general. In any theory with a preferred "universal rest frame" (e.g., Ni's theory; Box 39.1), there are additional terms in the metric due to motion of the sun relative to that preferred frame (exercise 40.1):

\begin{aligned} d s^{2} = & (expression 40.3) + (α_{2} + α_{3} - α_{1}) \frac{M_{⊙}}{r} w^{2} d t^{2} + 2 (α_{2} - \frac{1}{2} α_{1}) \frac{M_{⊙}}{r} w_{j} d x^{j} d t \\ (') & - α_{2} [\frac{M_{⊙}}{r^{3}} x^{j} x^{k} - \frac{I_{⊙}}{r^{5}} (x^{j} x^{k} - \frac{1}{3} r^{2} δ_{j k})] w_{j} d t (2 d x^{k} + w_{k} d t) . \end{aligned}

In these "preferred-frame terms"

I_{⊙} \equiv I_{j j} = \int ρ r^{2} d^{3} x

is the trace of the second moment of the sun's mass distribution;

\begin{aligned} α_{1} = 7 Δ_{1} + Δ_{2} - 4 γ - 4, \\ α_{2} = Δ_{2} + ζ - 1, \\ α_{3} = 4 β_{1} - 2 γ - 2 - ζ \end{aligned}

are combinations of PPN parameters; and

w

is the sun's velocity (

\equiv

velocity of coordinate system) relative to the preferred frame. (Theories such as general relativity and Dicke-Brans-Jordan, which possess no preferred frame, have

α_{1} = α_{2} = α_{3} = 0

, and therefore have no preferred-frame terms in the metric.) For ease of exposition, all equations and calculations in this chapter will ignore the preferred-frame terms; but the consequences of those terms will be discussed and references analyzing them will be cited.

Exercise 40.1. PPN METRIC FOR IDEALIZED SUN [Track 2]

Show that for an isolated, static, spherical sun at rest at the origin of the PPN coordinate system, the PPN metric (39.32) reduces to expressions (40.3), (40.3'). As part of the reduction, show that the sun's total mass-energy is given by

\begin{matrix} (40.4) & M_{⊙} = \int_{0}^{R_{⊙}} ρ_{0} (1 + 2 β_{2} U + β_{3} Π + 3 β_{4} p / ρ_{0}) 4 π r^{2} d r \end{matrix}

[Warning: One must not look at this formula and immediately think: "The contribution of rest mass is

\int ρ_{0} 4 π r^{2} d r

, the contribution of gravitational energy is

\int 2 β_{2} ρ_{0} U 4 π r^{2} d r

, etc." Rather, in making any such interpretation one must remember that (1) spacetime is curved, so

4 π r^{2} d r

is not proper volume as measured by physical meter sticks; also (2) virial theorems (exercise 39.6) and other integral theorems can be used to change the form of the integrand. For further discussion see exercises 40.9 and 40.10 below.]

For further discussion see exercises 40.9 and 40.10 below.]

EXERCISES

§40.2. THE USE OF LIGHT RAYS AND RADIO WAVES TO TEST GRAVITY

In the Newtonian limit, planetary and spacecraft orbits are strongly influenced by gravity; but light propagation and radio-wave propagation (at "infinite" velocity) are not influenced at all. For this reason, experimental studies of orbits are beset by the problem of separating the relativistic effects from much larger standard Newtonian effects. By contrast, experimental studies of light and radio-wave propagation do not contend with any such overpowering Newtonian background. Not surprisingly, they are to date (1973) the clearest and most definitive of the solarsystem experiments.

Mathematically, the parameter that distinguishes a light ray from a planet is its high speed. In the geodesic equation, the magnitude of the velocity determines which metric coefficients can influence the motion. Consider, for example, a weak, static field

g_{α β} = η_{α β} + h_{α β}

, and a particle at

(x, y, z) = (r, 0, 0)

moving with velocity

(v_{x}, v_{y}, v_{z}) = (0, v, 0)

; see Figure 40.1. Here the effect of gravity on the trajectory of the particle can be characterized by the quantity

(\binom{curvature of trajectory in 3-dimensional,}{nearly Euclidean, space}) = {(\binom{radius of curvature}{of trajectory})}^{- 1}

\begin{aligned} = \frac{d^{2} x}{d y^{2}} = \frac{d τ}{d y} \frac{d}{d τ} (\frac{d τ}{d y} \frac{d x}{d τ}) = \frac{1}{u^{y}} \frac{d}{d τ} (\frac{u^{x}}{u^{y}}) = \frac{1}{{(u^{y})}^{2}} \frac{d u^{x}}{d τ} \\ = - \frac{(1 - v^{2})}{v^{2}} Γ^{x}_{α β} \frac{d x^{α}}{d τ} \frac{d x^{β}}{d τ} = - \frac{1}{v^{2}} Γ^{x}_{α β} \frac{d x^{α}}{d t} \frac{d x^{β}}{d t} \\ = - Γ^{x}_{00} v^{- 2} - 2 Γ^{x}_{0 y} v^{- 1} - Γ^{x}_{y y} \\ = \frac{1}{2} h_{00, x} v^{- 2} + (h_{0 y, x} - h_{0 x, y}) v^{- 1} + (\frac{1}{2} h_{y y, x} - h_{x y, y}) \end{aligned}

Light rays and radio waves give "clean" tests of relativity

Figure 40.1.
The bending of the trajectory of a test body at its point of closest approach to the sun, as a function of its 3 -velocity. (See text for computation and discussion.)

Reexpressed in spherical coordinates, in the terminology of the idealized solar line element (40.3), this formula says

\begin{aligned} (\begin{array}{l} curvature of trajectory \\ in 3-space \end{array} & = \frac{1}{2} h_{00, r} v^{- 2} + \frac{1}{2} {(h_{ϕ ϕ} / r^{2})}_{, r} \\ \approx - (M_{⊙} / r^{2}) (v^{- 2} + γ) \end{aligned}

for a particle at its point of closest approach to the sun. (Compare with exercise 25.21.) Note that here

γ

is a PPN parameter; it is not

{(1 - v^{2})}^{- 1 / 2}

Notice what happens as one boosts the velocity of the particle. For slow velocities [

v^{2} \sim (

post-Newtonian expansion parameter

ϵ^{2}) \approx M_{⊙} / R_{⊙}]

, the Newtonian part of

h_{00}

dominates completely; and the tiny post-Newtonian corrections come equally from the

ϵ^{4}

part of

h_{00}

, the

ϵ^{3}

part of

h_{0 j}

, and the

ϵ^{2}

part of

h_{j k}

. [This was the justification for expanding

h_{00}

O (ϵ^{4}), h_{0 j}

O (ϵ^{3})

, and

h_{j k}

O (ϵ^{2})

in the postNewtonian limit; see §39.6.] But as

v

increases, the ordering of the terms changes. In the high-v regime

(v \sim 1 ≫ ϵ^{2})

, the bending of the trajectory has become almost imperceptible because of the high forward momentum of the particle and the short time it receives transverse momentum from the sun. What bending is left is due to the

ϵ^{2}

(Newtonian) part of

h_{00}

, and the

ϵ^{2}

(post-Newtonian) part of

h_{j k}

. Nothing else can have a significant influence. Notice, moreover, that-even when one allows for "preferred-frame" effects-these dominant terms,

h_{00} = 2 U = 2 M_{⊙} / r and h_{j k} = 2 γ U δ_{j k} = 2 γ (M_{⊙} / r) δ_{j k},

depend only on the Newtonian potential

U \equiv - Φ

and the PPN parameter

γ

.
This is a special case of a more general result: Aside from fractional corrections of

ϵ^{2} \leq 10^{- 6}

, relativistic effects on light and radio-wave propagation are governed entirely by the Newtonian potential

U

and the PPN parameter

γ

. These relativistic effects include the gravitational redshift (discussed in the last chapter; independent

Light rays are governed solely by Newtonian potential and PPN parameter

γ

γ

), the gravitational deflection of light and radio waves (discussed below; dependent on

γ

), and the "relativistic time-delay" (discussed below; dependent on

γ

§40.3. '"LIGHT"' DEFLECTION

Consider a light or radio ray coming into a telescope on Earth from a distant star or quasar. Do not assume, as in the usual discussion (exercises 18.6 and 25.24), that the ray passes near the sun. The deflection by the sun's gravitational field will probably be measurable, in the middle or late 1970 's, even when the ray passes far from the sun! [The calculation that follows is due to Ward (1970), but Shapiro (1967) first derived the answer.]

Orient the PPN spherical coordinates of equation (40.3) so that the ray lies in the "plane"

θ = π / 2

. By symmetry, if it starts out in this plane far from the Earth, it must lie in this plane always. Let the incoming ray enter the solar system along the line

ϕ = 0

; and let the Earth be located at

r = r_{E}, ϕ = ϕ_{E}

when the ray reaches it. (See Figure 40.2.) One wishes to calculate the angle

α

between the incoming light ray and the center of the sun, as measured in the orthonormal frame

(e_{\hat{r}}, e_{\hat{ϕ}})

of an observer on Earth. If the sun had zero mass (flat, Euclidean space),

α

would be

π - ϕ_{E}

(see Figure 40.2). However, the sun produces a deflection:

α = π - ϕ_{E}

+ δ α

. The deflection angle

δ α

is the true objective of the calculation.
In the calculation, ignore the Earth's orbital and rotational motions. They lead to aberration, for which correction can be made by the usual formula of special relativity (Lorentz transformation in the neighborhood of the telescope.) Also ignore deflection of the light ray due to the Earth's gravitational field (deflection angle ~

2 M_{E} / R_{E} \sim 0^{''} .0003

), which might be detectable in the late 1970's.

Figure 40.2.
Coordinates used in the text for calculating the deflection of light. Notice that in this diagram

ϕ

increases in the clockwise direction.

As the first step in calculating the deflection angle, determine the trajectory of the ray in the

r, ϕ

-plane. This can be calculated either using the geodesic equation, or using the eikonal method of geometric optics (Hamilton-Jacobi method; §22.5 and Box 25.4). The result of such a calculation (exercise 40.2) is an equation connecting

r

with

ϕ

; thus,

\begin{matrix} (40.6) & \frac{b}{r} = \sin ϕ + \frac{(1 + γ) M_{⊙}}{b} (1 - \cos ϕ) \end{matrix}

Notice that

b

has a simple geometric interpretation: far from the sun, the ray trajectory is

ϕ = b / r + O (M_{⊙} b / r^{2})

. Consequently,

b

is the impact parameter in the usual sense of classical scattering theory (see Figure 40.2). The ray makes its closest approach to the sun (assuming it is not intercepted by the Earth first) at the PPN coordinate radius

\begin{matrix} (40.7) & r_{min} = b [1 - \frac{(1 + γ) M_{⊙}}{b}] \approx b \end{matrix}

Thus,

b

can also be thought of as the radius of the ray's "perihelion."
Notice that the ray returns to

r = \infty

, not at an angle

ϕ = π

, but rather at

\begin{matrix} (40.8a) & ϕ (r = \infty) = π + 2 (1 + γ) M_{⊙} / b \end{matrix}

Thus, the total deflection angle is

\begin{aligned} (angle of total deflection) = 2 (1 + γ) M_{⊙} / b \\ (40.8b) & = \frac{1}{2} (1 + γ) 1^{''} .75 for a ray that \\ just grazes the sun. \end{aligned}

But this is not the quantity of primary interest. Rather, one seeks the position of the star as seen by an astronomer on Earth. The angle

α = π - ϕ_{E} + δ α

between the sun and the star as measured by the astronomer is given by (see Figure 40.2)

\begin{aligned} \tan (π - ϕ_{E} + δ α) & = - \tan ϕ_{E} + δ α / \cos^{2} ϕ_{E} \\ (40.9) & = \frac{u^{\hat{ϕ}}}{u^{r}} = {[\frac{(1 + γ M_{⊙} / r) r d ϕ / d λ}{(1 + γ M_{⊙} / r) d r / d λ}]}_{E} = {[\frac{r d ϕ}{d r}]}_{E} \\ = - {[\frac{(b / r) d ϕ}{d (b / r)}]}_{E} \end{aligned}

where

u^{β} = d x^{β} / d λ

are the components of a tangent to the ray at the Earth. By inserting into this equation expression (40.6) for the trajectory of the ray, one obtains

\begin{aligned} (40.10) & \tan ϕ_{E} - \frac{δ α}{\cos^{2} ϕ_{E}} & = \frac{\sin ϕ_{E} + [(1 + γ) M_{⊙} / b] (1 - \cos ϕ_{E})}{\cos ϕ_{E} + [(1 + γ) M_{⊙} / b] \sin ϕ_{E}} \\ = \tan ϕ_{E} - [(1 + γ) M_{⊙} / b] (1 - \cos ϕ_{E}) / \cos^{2} ϕ_{E} \end{aligned}

Thus, the deflection angle measured at the Earth is

\begin{matrix} (40.11) & δ α = \frac{(1 + γ) M_{⊙}}{b} (1 + \cos α) = \frac{(1 + γ) M_{⊙}}{r_{E}} {(\frac{1 + \cos α}{1 - \cos α})}^{1 / 2} \end{matrix}

It ranges from zero when the ray comes in opposite to the sun's direction

(α = π)

, through the value

\begin{matrix} (40.12) & (1 + γ) M_{⊙} / r_{E} = \frac{1}{2} (1 + γ) 0^{''} .0041 \end{matrix}

when the ray comes in perpendicular to the Earth-Sun line (

α = π / 2

), to the "classical value" of

\frac{1}{2} (1 + γ) \times 1^{''} .75

when the ray comes in grazing the sun's limb.

All experiments to date (1972) have examined the case of grazing passage. The experimental results are stated and discussed in Box 40.1. They show that the PPN parameter

γ

has its general relativistic value of 1 to within an uncertainty of about 20 percent.

By the middle or late 1970's, measurements of the deflection of radio waves from quasars should determine

γ

to much better than 1 percent. Also, by that time radio astronomers may be making progress toward setting up high-precision coordinates on the sky using very long baseline interferometry. If so, they will have to use equation (40.11) to compensate for the "warping" of the coordinates caused by the sun's deflection of radio waves in all regions of the sky, not just near the solar limb.
(2) formula for deflection angle

Experimental measurements of deflection

Exercise 40.2. TRAJECTORY OF LIGHT RAY IN SUN'S GRAVITATIONAL FIELD

Derive equation (40.6) for the path of a light ray in isotropic coordinates (40.3) in the sun's "equatorial plane." Use one or more of three alternative approaches: (1) direct integration of the geodesic equation (the hardest approach!); (2) computation based on the three integrals of the motion

\begin{matrix} k \cdot k = 0, k \cdot (\partial / \partial t) = k_{0}, k \cdot (\partial / \partial ϕ) = k_{ϕ} = - b k_{0} \\ k \equiv d / d λ = tangent vector to geodesic \end{matrix}

(see

§ § 25.2

and 25.3); (3) computation based on the Hamilton-Jacobi method (Box 25.4), which for photons (zero rest mass) reduces to the "eikonal method" of geometric optics (see §22.5).

EXERCISE

§40.4. TIME-DELAY IN RADAR PROPAGATION

Another effect of spacetime curvature on electromagnetic waves is a relativistic delay in the round-trip travel time for radar signals. It was first pointed out by Shapiro (1964); see also Muhleman and Reichley

(1964, 1965)

Eclipse Measurements

Until 1968 every experiment measured the deflection of starlight during total eclipse of the sun. The measurements were beset by difficulties such as poor weather, optical distortions due to temperature changes, and the strange propensity of eclipses to attain maximum time of totality in jungles, in the middles of oceans, in deserts, and in arctic tundras. Lists of all the results and references are given by Bertotti, Brill and Krotkov (1962), and by Klüber (1960). Dicke (1964b) summarizes the results as follows:
"The analyses [of the experimental data] scatter from a deflection at the limb of the sun of 1.43 seconds of arc to 2.7 seconds [compared to a general relativistic value of 1.75 seconds]. The scatter would not be too bad if one could believe that the technique was free of systematic errors. It appears that one must consider this observation uncertain to at least 10 percent, and perhaps as much as 20 percent." This result corresponds to an uncertainty in

γ

of 20 to 40 percent.

Measurements on the Deflection of Radio Waves

Each October 8 the sun, as seen from the Earth, passes in front of the quasar 3C279. By monitoring the angular separation between 3C279 and a nearby quasar 3C273, radio astronomers can measure the deflection by the sun of the 3C279 radio waves. The monitoring uses radio interferometers. [See references cited in table for discussion of the technique.] Technology of the early 1970's should permit measurements to a precision 0.001 seconds of arc or better, if the two ends of the interferometer are separated by several thousand kilometers ("transcontinental" or "transworld" baseline). But as of 1971 the only successful experiments were less ambitious: they used baselines of less than 10 kilometers. A summary of these pre-1971, short-baseline results is shown in the table.

The 90 -foot (background) and 130 -foot (foreground) radio interferometer system at Caltech's Owens Valley Radio Observatory. These were used by the deflection of quasar radio waves by the sun. During the experiment the two antennas were separated by 1.07 kilometers. (Photo by Alan Moffet.)

Dates of observation

Observatory

Experimenters and reference

Number of telescopes and separations

Wave lengths

Experimental results

^{a}

\frac{1}{2} (1 + γ) = \frac{(\binom{Observed}{deflection})}{(\binom{Einstein}{prediction})}

Formal standard error

Onesigma error

\begin{matrix} Sept. 30 - Oct. 15 \\ 1969 \end{matrix}

Owens Valley (Caltech)

Seielstadt, Sramek, Weiler (1970)

\begin{matrix} 2, \\ 1.07 km \end{matrix}

3.1 cm

1.01

\pm 0.12

\pm 0.12

\begin{aligned} Oct. 2-Oct. 10 \\ 1969 \end{aligned}

Goldstone (Caltech-JPL)

Muhleman, Ekers,

Fomalont (1970)

\begin{matrix} 2, \\ 21.56 km \end{matrix}

12.5 cm

1.04

\pm 0.05

\begin{aligned} + 0.15 \\ - 0.10 \end{aligned}

\begin{matrix} Oct. 2 -Oct. 12 \\ 1970 \end{matrix}

National Radio Astronomy Observatory (USA)

Sramek (1971)

\begin{matrix} 3, \\ 0.80 km, 1.90 km, \\ 2.70 km \end{matrix}

\begin{array}{r} 11.1 cm \\ 3.7 cm \end{array}

0.90

\pm 0.05

\pm 0.05

\begin{matrix} Sept. 30 - Oct. 15 \\ 1970 \end{matrix}

Mullard Radio

Astronomy

Observatory (Cambridge Univ.)

Hill (1971)

\begin{matrix} 3, \\ 0.66 km, 1.41 km \end{matrix}

\begin{array}{r} 11.1 cm \\ 6.0 cm \end{array}

1.07

\pm 0.17

\pm 0.17

^{a}

Here (observed deflection)/(Einstein prediction) is the number

\frac{3}{2} (1 + γ)

obtained by fitting the observational data to the PPN prediction (40.11). [For these experiments the ray passes near the solar limb; so (40.11) reduces to

δ α = \frac{1}{2} (1 + γ) (M_{⊙} / b)

.] The "formal standard error" is obtained from the data by standard statistical techniques. However, it is not usually a good measure of the certainty of the result, because it fails to take account of systematic errors. The quoted "one-sigma error" is the experimenters' best estimate of the combined statistical and systematic
uncertainties. The experimenters estimate a probability of 68 percent that the true result is within "lo" of their measured value; a probability of 95 percent that it is within "

2 σ

"; etc.
(1) foundations for calculation; Fermat's principle
(2) details of calculation

Figure 40.3.
Diagram, in the PPN coordinate system, for the calculation of the relativistic time delay.

Let a radar transmitter on Earth send a radio wave out to a reflector elsewhere in the solar system, and let the reflector return the wave to Earth. Calculate the round-trip travel time, as measured by a clock on Earth. For simplicity of calculation, idealize both Earth and reflector as nonrotating and as at rest in the static, spherical gravitational field of the Sun. At the end of the calculation, the effects of rotation and motion will be discussed separately. Also ignore time dilation of the transmitter's clock due to the Earth's gravitational field; it is easily corrected for, and it is so small that it will not come into play in these radar experiments before the middle or late 1970's. The gravitational effects of the other planets on the radio waves are too small to be discernible in the foreseeable future, unless the beam grazes the limb of one of them. However, the effects of dispersion in the solar wind and corona are discernible and must be corrected for. These corrections will not be discussed here, since they are free of any general-relativistic influence.

The calculation of the round-trip travel time can be simplified by using a generalrelativistic version of Fermat's principle: In any static field (

g_{0 j} = 0, g_{α β, 0} = 0

) consider all null curves between two points in space,

x^{j} = a^{j}

and

x^{j} = b^{j}

. Each such null curve,

x^{j} (t)

, requires a particular coordinate time interval

Δ t

to get from

a^{j}

b^{j}

. The curves of extremal

Δ t

are the null geodesics of spacetime. The proof of this theorem is outlined in exercise 40.3.

Because of Fermat's principle, the lapse of coordinate time between transmission of the radar beam and reflection at the reflector,

t_{TR}

, is the same for a straight path in the PPN coordinates as for the slightly curved path which the beam actually follows. (The two differ by a fractional amount

Δ t_{TR} / t_{TR} \sim

(angle of deflection)

^{2} ≦ 10^{- 12}

, which is far from discernable.) Hence, in the computation one can ignore the gravitational bending of the beam.

Adopt Cartesian PPN coordinates with the sun at the origin; the transmitter, sun, and reflector in the

z = 0

"plane"; and the transmitter-reflector line along the

x

direction (see Figure 40.3). The transmitter is at

(x, y) = (- a_{T}, b)

in the PPN coordinates, and the reflector is at

(x, y) = (a_{R}, b)

. Recall that for a null ray

d s^{2} = 0 =

g_{00} d t^{2} - g_{x x} d x^{2}

. It follows that the lapse of coordinate time between transmission and reflection is

t_{TR} = \int_{- a_{T}}^{a_{R}} {(\frac{g_{x x}}{- g_{00}})}^{1 / 2} d x = \int_{- a_{T}}^{a_{R}} [1 + \frac{(1 + γ) M_{⊙}}{\sqrt{x^{2} + b^{2}}}] d x

Integration yields

\begin{matrix} (40.13) & t_{TR} = a_{R} + a_{T} + (1 + γ) M_{⊙} \ln [\frac{(a_{R} + \sqrt{a_{R}^{2} + b^{2}}) (a_{T} + \sqrt{a_{T}^{2} + b^{2}})}{b^{2}}] . \end{matrix}

The lapse of coordinate time in round-trip travel has twice this magnitude. The lapse of proper time measured by an Earth-based clock is

\begin{aligned} Δ τ = & {| g_{00} |}_{Earth} 2 t_{TR}, \\ Δ τ = & 2 (a_{R} + a_{T}) (1 - \frac{M_{⊙}}{\sqrt{a_{T}^{2} + b^{2}}}) \\ (40.14) & + 2 (1 + γ) M_{⊙} \ln [\frac{(a_{R} + \sqrt{a_{R}^{2} + b^{2}}) (a_{T} + \sqrt{a_{T}^{2} + b^{2}})}{b^{2}}] \end{aligned}

(3) formula for delay

This is the lapse of time on an Earth-based clock, aside from corrections for the orbital and rotational motion of the Earth, for the orbital motion of the reflector, for dispersion of radiation traversing the solar wind and corona, and for time dilation in the Earth's gravitational field.

Any reader is reasonable who objects to the form (40.14) in which the time-delay has been written. The quantities

a_{R}, a_{T}

, and

b

are coordinate positions in the PPN coordinate system, rather than numbers the astronomer can measure directly. They differ from coordinate positions in other, equally good coordinate systems by amounts of the order of

M_{⊙} \sim 1.5 km

. The objection is not mathematical in its origin. The quantities

a_{R}, a_{T}

, and

b

are perfectly well-defined [with post-post-Newtonian uncertainties of order

b {(M_{⊙} / b)}^{2} ≦ 10^{- 6} km]

, because the PPN coordinate system is perfectly well-defined. But they are not quantities which the experimenter can measure directly, with precision anywhere near that required to test the relativistic terms in the time-delay formula (40.14).

In practice, fortunately, the experimenter does not need to measure

a_{R}, a_{T}

, or

b

with high precision. Instead, he checks the time-delay formula by measuring the changes in

Δ τ

as the Earth and reflector move in their orbits about the Sun; i.e., he measures

Δ τ

as a function of Earth-based time

τ

. Notice that when the beam is passing near the sun

(b ≪ a_{R}, b ≪ a_{T}

; but

d b / d τ ≫ d a_{R} / d τ

and

d b / d τ ≫ d a_{T} / d τ

because the Earth's and reflector's orbits are nearly circular), the change of

b

in the

\ln

term of (40.14) dominates all other relativistic corrections to the Newtonian delay; consequently (using

d b / d τ \sim 10 km / \sec

for typical experiments)

\begin{aligned} (40.15) & \frac{d Δ τ}{d τ} - (\binom{Constant Newtonian}{part}) \approx - 4 (1 & + γ) \frac{M_{⊙}}{b} \frac{d b}{d τ} \\ \sim 4 (1 + γ) (\frac{1.5 km}{10^{6} km}) (\frac{10 km}{\sec}) \sim \frac{30 μ \sec}{day} \end{aligned}

(4) comparison with experiment

Such differential shifts in round-trip travel time-which rise as the Earth-reflector line moves toward the Sun and falls as it moves away-are readily observable.

In practice, in order to obtain precisions better than about 20 percent in the determination of the parameter

γ

by time-delay measurements, one must carefully collect and analyze data for a large fraction of a year-from a time when the beam is far from the sun

(b \sim a_{T} \sim 10^{8} km)

, to the time of superior conjunction (

b \sim R_{⊙}

\sim 10^{6} km

), and on around to a time of distant beam again. Such a long "arc" of data is needed to determine the reflector's orbit with high precision, and to take full advantage of the slow, logarithmic falloff of

Δ τ

with

b

(40.14). When the beam is far from the sun

(b ≫ R_{⊙})

, the simplifying assumptions behind equation (40.15) are not valid; and the relativistic time-delay gets intertwined with the orbital motions of the Earth and the reflector. The analysis then remains straightforward, but its details are so complex that one resorts to numerical integrations on a computer to carry it out. Because the orbital motions enter, the time-delay data then contain information about other metric parameters (

β

is the dominant one) in addition to

γ

The experimental results as of 1971 are described in Box 40.2. They yield a value for the PPN parameter

γ

that is more accurate than the value from light and radio-wave deflection experiments:

\begin{matrix} (40.16) & γ = 1.02 \pm 0.08 \end{matrix}

Future experiments using spacecraft may improve the precision of

γ

\pm 0.001

or better.

EXERCISE

Exercise 40.3. FERMAT'S PRINCIPLE

Prove Fermat's principle for a static gravitational field. [Hint: The proof might proceed as follows. Write down the geodesic equation in four-dimensional spacetime using an affine parameter

λ

. Convert from the parameter

λ

to coordinate time

t

, and use

d s^{2} = 0

to obtain

g_{j k} \frac{d^{2} x^{k}}{d t^{2}} + Γ_{j k ℓ} \frac{d x^{k}}{d t} \frac{d x^{ℓ}}{d t} - Γ_{j 00} \frac{g_{k ℓ}}{g_{00}} \frac{d x^{k}}{d t} \frac{d x^{ℓ}}{d t} + \frac{d^{2} t / d λ^{2}}{(d t / d λ)^{2}} g_{j k} \frac{d x^{k}}{d t} = 0

Combine with the time part of the geodesic equation

\frac{d^{2} t / d λ^{2}}{(d t / d λ)^{2}} = - 2 Γ_{0 k 0} \frac{d x^{k} / d t}{g_{00}}

and use the expression for the Christoffel symbols in terms of the metric to obtain

γ_{j k} \frac{d^{2} x^{k}}{d t^{2}} + \frac{1}{2} (γ_{j k, ℓ} + γ_{j l, k} - γ_{k l, j}) \frac{d x^{k}}{d t} \frac{d x^{ℓ}}{d t} = 0, γ_{j k} \equiv - \frac{g_{j k}}{g_{00}}

Then notice that this is a geodesic equation with affine parameter

t

in a three-dimensional manifold with metric

γ_{j k}

. The familiar extremum principle for this geodesic is

δ \int_{a^{j}}^{b^{j}} {(γ_{j k} d x^{j} d x^{k})}^{1 / 2} = δ \int_{a^{j}}^{b^{i}} d t = 0,

which is precisely Fermat's principle!]

Box 40.2 RADAR TIME DELAY IN THE SOLAR SYSTEM: EXPERIMENTAL RESULTS

Two types of experiments have been performed to measure the relativistic effects [proportional to

\frac{1}{2} (1 + γ)

; equation (40.14)] in the round-trip radar travel time in the solar system. In one type ("passive" experiment) the reflector is the surface of the planet Venus or the planet Mercury. In the other type ("active" experiment) the "reflector" is electronic equipment on board a spacecraft that receives the signal and transmits it back to Earth ("transponder"). Passive experiments suffer from noise due to topography of the reflecting planet (earlier radar return from mountain tops than from valley floors), and they suffer from weakness of the returned signal. Active experiments suffer from buffeting of the spacecraft by solar wind, buffeting by fluctuations in solar radiation pressure, and buffeting by leakage from gas jets ("outgassing"). Experiments of the future will solve these problems by placing a transponder on the surface of a planet or on a "drag-free" (buffetingfree) spacecraft. But experiments of the present and future must both contend with fluctuating time delays due to dispersion in the fluctuating solar wind and corona. Fortunately, these are smaller than the relativistic effects, except when

The Mariner VI spacecraft (mock-up), which was the reflector in a 1970 measurement of

\frac{1}{2} (1 + γ)

by radar time delay [photo courtesy the Caltech Jet Propulsion Laboratory].
the beam passes within 2 or 3 solar radii of the sun.
The results of experiments performed before 1972 are listed in the table.

Dates of observation

Radar telescopes

Reflector

Experimenters

and reference

Experimental result

^{a}

\frac{1}{2} (1 + γ) =

Formal standard error

Onesigma error

Wave length

\frac{(\binom{Observed}{delay})}{(\binom{Einstein}{prediction})}

\begin{aligned} November 1966 \\ to \end{aligned}

Haystack (MIT)

Venus and Mercury

Shapiro (1968)

3.8 cm

0.9

\pm 0.2

August 1967

1967

through

Haystack (MIT), and

Venus and Mercury

Shapiro, Ash, et al. (1971)

3.8 cm

and

1.015

\pm 0.02

\pm 0.05

through

1970

and

Arecibo (Cornell)

Mercury

(1971)

\begin{matrix} and \\ 70 cm . \end{matrix}

October 1969

Deep Space

Mariner VI

Anderson, et al.

14 cm .

1.00

\pm 0.014

\pm 0.04

Network

and VII

(1971)

January 1971

(NASA)

spacecraft

§40.5. PERIHELION SHIFT AND PERIODIC PERTURBATIONS IN GEODESIC ORBITS

Perihelion shift for geodesic orbits around spherical sun, ignoring preferred-frame effects

The light-deflection and time-delay experiments both measured

γ

. To measure other PPN parameters, one must examine the effects of gravity on slowly moving bodies; this was the message of

§ 40.2

Begin with the simplest of cases: the geodesic orbit of a test body in the sun's spherical gravitational field, ignoring all gravitational effects of the planets, of solar oblateness, and of motion relative to any preferred frame. The PPN metric then has the form (40.3):

\begin{aligned} d s^{2} = & - [1 - 2 \frac{M_{⊙}}{r} + 2 β \frac{M_{⊙}^{2}}{r^{2}}] d t^{2} \\ (40.3) & + [1 + 2 γ \frac{M_{⊙}}{r}] [d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})] \end{aligned}

Orient the coordinates so the test body moves in the equatorial "plane"

θ = π / 2

; and calculate the shape

r (ϕ)

of its nearly Keplerian, nearly elliptical geodesic orbit. The result, accurate to order

M_{⊙} / r

beyond Newtonian theory, is

\begin{matrix} (40.17) & r = \frac{(1 - e^{2}) a}{1 + e \cos [(1 - δ ϕ_{0} / 2 π) ϕ]} \end{matrix}

where

a

and

e

are constants of integration, and

δ ϕ_{0}

is defined by

\begin{aligned} δ ϕ_{0} & = \frac{(2 - β + 2 γ)}{3} \frac{6 π M_{⊙}}{a (1 - e^{2})} \\ (40.18) & = \frac{6 π M_{⊙}}{a (1 - e^{2})} in general relativity. \end{aligned}

(For derivation, see exercise 40.4.)
Notice that, if

δ ϕ_{0}

were zero-as it is in the Newtonian limit-then the orbit (40.18) would be an ellipse with semimajor axis

a

and eccentricity

e

(see Box 25.4). The constant

δ ϕ_{0}

merely makes the ellipse precess: for

r

to go through a complete circuit, from perihelion to aphelion to perihelion again,

(1 - δ ϕ_{0} / 2 π) ϕ

must change by

2 π

; so

ϕ

must change by

2 π + δ {\dot{ϕ}}_{0}

. Thus, the perihelion shifts forward by an angle

δ ϕ_{0}

with each circuit around the ellipse.

Relative to what does the perihelion shift? (1) Relative to the PPN coordinate system; (2) relative to inertial frames at the outskirts of the solar system (since the PPN coordinates are tied to those frames; see

§ 39.12

); (3) relative to a frame determined by the "fixed stars in the sky" (since the inertial frames at the outskirts of the solar system, inertial frames elsewhere in our galaxy, and inertial frames in our cluster of galaxies should not rotate significantly relative to each other); (4) relative to the perihelia of (other) planets, which themselves are shifting at calculable rates that decrease as one moves outward in the solar system from Mercury to Venus to Earth to....

The perihelion shift is not the only relativistic effect contained in the orbital motion for a test body. There are other effects, but they are all periodic rather than cumulative with time; so, with the limited technology of the pre-space-age era, it was impossible to detect them. But the technology of the 1970's is bringing them within reach. Moreover, many space-age experiments are necessarily of short duration (

£

one orbit)-particularly those involving spacecraft and transponders landed on planets. For these, the periodic perturbations in an orbit are of almost as much experimental value as the cumulative perihelion shift. The periodic effects are not obvious in the PPN orbital equation (40.17); it looks like the simplest of precessing ellipses. But the quantities the observer measures directly are not

a, e

, and

δ ϕ_{0}

. Instead, he measures the time evolution of round-trip radar travel times,

Δ τ (τ)

, and of angular positions on the sky

[θ_{0} (τ), ϕ_{0} (τ)]

. To compute these quantities is perfectly straightforward in principle, but in practice is a very complex task. The calculations predict relativistic effects periodic with the frequency of the orbit and all its harmonics. The amplitudes of these effects, for the lower harmonics, must obviously be of the order of

M_{⊙} \sim 1 km \sim 10 μ \sec \sim 0^{''} .01

arc on the sky. (The distance

M_{⊙} = 1.48 km

is the characteristic length for all relativistic effects in the sun's spherical field!)

The most favorable orbits for experimental tests of the perihelion shift and of periodic effects are those that go nearest the sun and have the highest eccentricity [see equation (40.18)]-the orbits of Mercury, Venus, Earth, Mars, and the asteroid Icarus. But how does one know that these orbits are geodesics? After all, planets are not "test bodies"; they themselves produce nonnegligible curvature in spacetime. It turns out (see

§ 40.9

for full discussion) that there should exist tiny deviations from geodesic motion, but they are too small to compete with the perihelion shift or with the periodic effects discussed above, at least for these five bodies.

Extensive astronomical observations of planetary orbits, dating back to the midnineteenth century and aided by radar since 1966, have yielded accurate values for planetary perihelion shifts (accurate to

\pm 0.4

seconds of arc per century for Mercury). From the data, which are summarized and discussed in Box 40.3, one obtains the value

\begin{matrix} (40.19a) & \frac{1}{3} (2 - β + 2 γ) = 1.00 {\begin{matrix} + 0.01 \\ - 0.10 \end{matrix} \end{matrix}

for the ratio of observed relativistic shift to general relativistic prediction. Combining this result with the radar-delay value for

γ

(40.16), one obtains a value

\begin{matrix} (40.19b) & β = 1.0 {\begin{matrix} + 0.4 \\ - 0.2 \end{matrix} \end{matrix}

for the PPN parameter

β

. (Recall:

β

measures the "amount of nonlinearity in the superposition law for

g_{00}

.")

The periodic effects in the planetary orbits have not yet (1973) been studied experimentally.

The above discussion and Box 40.3 have ignored the motion of the solar system relative to the preferred frame (if one exists); i.e., they have ignored the terms (40.3')

Periodic perturbations in geodesic orbits

Comparison of theory with planetary orbits

Experimental result for

β

Box 40.3 PERIHELION SHIFTS; EXPERIMENTAL RESULTS

Relativistic corrections to Newtonian theory are not the only cause of shift in the perihelion of a planetary orbit. Any departure of the Newtonian gravitational field from its idealized, spherical, inverse-square-law form also produces a shift. Such nonsphericities and resulting shifts are brought about by (1) the gravitational pulls of other planets, and (2) deformation of the sun ("solar oblateness"; "quadrupole moment"). In addition, when the primary data are optical positions of planets on the sky (right ascension and declination as functions of time), there is an apparent perihelion shift caused by the precession of the Earth's axis ("general precession"; observer not on a "stable platform"; see exercise 16.4).
The perihelion shifts due to a general precession and to the gravitational pulls of other planets can be calculated with high precision. But in 1973 there is no fully reliable way to determine the solar quadrupole moment. It is conventional to quantify the sun's quadrupole moment by a dimensionless parameter

J_{2}

, which appears in the following expression for the Newtonian potential,

U = \frac{M_{⊙}}{r} [1 - J_{2} \frac{R_{⊙}^{2}}{r^{2}} (\frac{3 \cos^{2} θ - 1}{2})] .

If the sun were rotating near breakup velocity,

J_{2}

would be near 1 . Very careful measurements of the optical shape of the sun [Dicke and Goldenberg (1967)] show a flattening, which suggests

J_{2}

may be near

3 \times 10^{- 5}

.
The total perihelion shift produced by relativity plus solar quadrupole moment is (see exercise 40.5)

\begin{aligned} δ ϕ & = \frac{6 π M_{⊙}}{a (1 - e^{2})} λ_{p}, \\ λ_{p} & \equiv \frac{2 - β + 2 γ}{3} + J_{2} \frac{R_{⊙}^{2} / M_{⊙}}{2 a (1 - e^{2})} . \end{aligned}

The Haystack radar antenna, which Irwin Shapiro and his group have used to collect extensive data on the systematics of the inner part of the solar system. Those data are rapidly becoming the most important source of information about perihelion shifts. (Picture courtesy of Lincoln Laboratories, MIT.)

Note that relativistic and quadrupole shifts have different dependences on the semimajor axis

a

and ecentricity

e

of the orbit. This difference in dependence allows one to obtain values for both the quadrupole moment parameter

J_{2}

, and the PPN parameter

\frac{1}{3} (2 - β + 2 γ)

by combining measurements of

δ ϕ

for more than one planet.

The experimental results, as of 1972 , are as follows.
I. Data for Mercury from optical studies [Clemence (1943, 1947)]

^{*}

(general relativity with no solar oblateness predicts

{43.03}^{''} /

century)

Quantity	Value
(a) Total observed shift per century	$5599 " .74 \pm 0^{''} .41$
(b) Contribution to shift caused by observer not being in an inertial frame far from
the sun ("general precession" as evaluated in 1947)	$5025^{''} .645 \pm 0^{''} .50$
(c) Shift per century produced by Newtonian gravitation of other planets	$531 " .54 \pm 0 " .68$
(d) Residual shift per century to be attributed to general relativity plus
solar oblateness	$42^{''} .56 \pm 0^{''} .94$
(e) Residual shift if one uses the 1973 value for the "general precession"	$41^{''} .4 \pm 0^{''} .90$
(f) Corresponding value of $λ_{p}$ (see above)	$λ_{p} = 0.96 \pm 0.02$

II. 1970 Results of Shapiro (1970, 1971a,b), Shapiro et al. (1972)
(a) Values of

λ_{p}

obtained by reanalyzing all the world's collection of optical data, and combining it with radar data
(b) Value of

J_{2}

obtained by comparing the observed shifts for Mercury and Mars

{\begin{cases} {(λ_{p})}_{Mercury} = 1.00 \pm 0.01 \\ {(λ_{p})}_{Mars} = 1.07 \pm 0.10 \end{cases}

J_{2} ≦ 3 \times 10^{- 5}

III. Theoretical implications of Shapiro's results
(a) Value of

(2 - β + 2 γ) / 3

1.00 {\begin{cases} + 0.01 \\ - 0.10 \end{cases}

(b) Value of

β

obtained by combining with

γ

from time delay experiments [equation (40.16)]

1.0 {\begin{cases} + 0.4 \\ - 0.2 \end{cases}

*Clemence (1947) notes, "The observations cannot be made in a Newtonian frame of reference. They are referred to the moving equinox, that is, they are affected by the precession of the equinoxes, and the determination of the precessional motion is one of the most difficult problems of observational astronomy, if not the most difficult. In the light of all these hazards, it is not surprising that a difference of opinion could exist regarding the closeness of agreement between the observed and theoretical motions."
in the sun's metric. When one takes account of these terms, one finds an additional contribution to the perihelion shift, given for small eccentricities

e ≪ 1

\begin{aligned} (40.20) & δ ϕ_{0} = & - α_{1} \frac{π}{2 e} {(\frac{M_{⊙}}{a})}^{1 / 2} w \cdot Q - α_{2} \frac{π}{4} [(w \cdot P)^{2} - (w \cdot Q)^{2}] \\ + α_{3} \frac{π}{e} (\frac{| Ω_{⊙} |}{M_{⊙}}) (\frac{ω_{⊙} a^{2}}{M_{⊙}}) w \cdot Q \end{aligned}

[see Nordtvedt and Will (1972)]. Here

M_{⊙}, Ω_{⊙}

, and

ω_{⊙}

are the sun's mass, self-gravitational energy, and rotational angular velocity;

w

is the sun's velocity relative to the preferred frame;

a

and

e

are the semimajor axis and eccentricity of the orbit;

P

is the unit vector pointing from the sun to the perihelion; and

Q

is a unit vector orthogonal to

P

and lying in the orbital plane. Comparison with observations for

Perihelion shift due to preferred-frame forces

Mercury-and combination with limits on

α_{1}

and

α_{2}

discussed below [equations (40.46b) and (40.48)]-yields the limit

\begin{matrix} (40.21a) & | α_{3} \frac{w \cdot Q}{200 km / \sec} | ≦ 2 \times 10^{- 5} . \end{matrix}

Since the velocity of the sun around the Galaxy is

\sim 200 km / \sec

, and the peculiar motion of the Galaxy relative to other nearby galaxies is

\sim 200 km / \sec

, a value

w \sim 200 km / \sec

is reasonable. Moreover, there is no reason to believe that

w

and

Q

are orthogonal, so one is fairly safe in concluding

\begin{matrix} (40.21b) & | α_{3} | = | 4 β_{1} - 2 γ - 2 - ζ | ≲ 2 \times 10^{- 5} \end{matrix}

This is a stringent limit on theories that possess universal rest frames. For example, with great certainty it rules out a theory devised by Coleman (1971), which has

β = γ = 1

, but

α_{3} = - 4

; see Ni (1972).

Looking toward the future, one cannot expect data on orbits of spacecraft to give decisive tests of general relativity, despite the high precision (

\sim 10

meters in 1972) with which spacecraft can be tracked. Spacecraft are buffeted by the solar wind. They respond to fluctuations in this wind and in the pressure of solar radiation, and respond also to "outgassing" from leaky jets. Unless one can develop excellent "drag-free" or "conscience-guided" spacecraft, one must therefore continue to rely on planets as the source of data on geodesics. However, planetary data themselves can be greatly improved in the future by placing radar transponders on the surfaces of planets or in orbit about them, by improvements in radar technology, and by the continued accumulation of more and more observations.

EXERCISES

Exercise 40.4. DERIVATION OF PERIHELION SHIFT IN PPN FORMALISM

[See exercise 25.16 for a derivation in general relativity, accurate when gravity is strong (

2 M / r

as large as

\frac{1}{3}

) but the orbital eccentricity is small. The present exercise applies to any "metric theory" and to any eccentricity, but it assumes gravity is weak (

2 M / r ≪ 1

) and ignores motion relative to any universal rest frame.] Derive equation (40.17) for the shape of any bound orbit of a test particle moving in the equatorial plane of the PPN gravitational field (40.3). Keep only "first-order" corrections beyond Newtonian theory (first order in powers of

M_{⊙} / r

). [Sketch of solution using Hamilton-Jacobi theory (Box 25.4): (1) Hamilton-Jacobi equation, referred to a test body of unit mass, is

\begin{aligned} - 1 & = g^{α β} {\tilde{S}}_{, α} {\tilde{S}}_{, β} \\ = - [1 + 2 \frac{M_{⊙}}{r} + (4 - 2 β) {(\frac{M_{⊙}}{r})}^{2}] {(\frac{\partial \tilde{S}}{\partial t})}^{2} + [1 - 2 γ \frac{M_{⊙}}{r}] [{(\frac{\partial \tilde{S}}{\partial r})}^{2} + \frac{1}{r^{2}} {(\frac{\partial \tilde{S}}{\partial ϕ})}^{2}] \end{aligned}

(2) Solution to Hamilton-Jacobi equation is

\begin{aligned} (40.22) & \tilde{S} = & - \tilde{E} t + \tilde{L} ϕ \pm \int^{r} {- (1 - {\tilde{E}}^{2}) + \frac{2 M_{⊙}}{r} [1 - (1 + γ) (1 - {\tilde{E}}^{2})] \\ {- \frac{{\tilde{L}}^{2}}{r^{2}} [1 - \frac{2 M_{⊙}^{2}}{{\tilde{L}}^{2}} (2 - β + 2 γ)]}}^{1 / 2} d r \end{aligned}

where post-post-Newtonian corrections have been discarded. In discarding post-post-Newtonian corrections, recall that

\tilde{E}

is the conserved energy per unit rest mass and

\tilde{L}

is the angular momentum per unit rest mass (see Box 25.4). Consequently one has the order-of-magnitude relations

1 - {\tilde{E}}^{2} \sim (velocity of test body)^{2} \sim M_{⊙} / r

and

{(M_{⊙} / \tilde{L})}^{2} \sim {(M_{⊙} / r v)}^{2} \sim M_{⊙} / r

(3) The shape of the orbit is determined by the "condition of constructive interference,"

\partial \tilde{S} / \partial \tilde{L} = 0 :

\begin{aligned} ϕ = \pm \int { & - \frac{1 - {\tilde{E}}^{2}}{{\tilde{L}}^{2}} + \frac{2 M_{⊙}}{{\tilde{L}}^{2} r} [1 - (1 + γ) (1 - {\tilde{E}}^{2})] \\ {- \frac{1}{r^{2}} [1 - \frac{2 M_{⊙}^{2}}{{\tilde{L}}^{2}} (2 - β + 2 γ)]}}^{- 1 / 2} d (1 / r) \end{aligned}

(4) This integral is readily evaluated in terms of trigonometric functions. For a bound orbit (

\tilde{E} < 1

), the integral is

ϕ = (1 + \frac{δ ϕ_{0}}{2 π}) \cos^{- 1} [\frac{(1 - e^{2}) a}{e r} - \frac{1}{e}]

where

\begin{matrix} a \equiv \frac{M_{⊙}}{1 - {\tilde{E}}^{2}} [1 - (1 + γ) (1 - {\tilde{E}}^{2})] \\ (40.23) & 1 - e^{2} \equiv {(\frac{\tilde{L}}{M_{⊙}})}^{2} (1 - {\tilde{E}}^{2}) [1 + 2 (1 + γ) (1 - {\tilde{E}}^{2}) - 2 {(\frac{M_{⊙}}{\tilde{L}})}^{2} (2 - β + 2 γ)] \\ δ ϕ_{0} \equiv \frac{1}{3} (2 - β + 2 γ) 6 π {(M_{⊙} / \tilde{L})}^{2} \end{matrix}

(5) Straightforward manipulations bring this result into the form of equations (40.17) and (40.18).]

Exercise 40.5. PERIHELION SHIFT FOR OBLATE SUN

(a) The Newtonian potential for an oblate sun has the form

\begin{matrix} (40.24) & U = \frac{M_{⊙}}{r} (1 - J_{2} \frac{R_{⊙}^{2}}{r^{2}} \frac{3 \cos^{2} θ - 1}{2}) \end{matrix}

where

J_{2}

is the "quadrupole-moment parameter." One knows that

J_{2} ≦ 3 \times 10^{- 5}

. Show that if an oblate sun is at rest at the origin of the PPN coordinate system, the metric of the surrounding spacetime [equations (39.32)] can be put into the form

\begin{aligned} d s^{2} = & - [1 - 2 \frac{M_{⊙}}{r} - 2 J_{2} (\frac{M_{⊙} R_{⊙}^{2}}{r^{3}}) (\frac{3 \cos^{2} θ - 1}{2}) + 2 β {(\frac{M_{⊙}}{r})}^{2}] d t^{2} \\ (40.25) & + [1 + 2 γ \frac{M_{⊙}}{r}] [d r^{2} + r^{2} (d θ^{2} + \sin^{2} θ d ϕ^{2})] \end{aligned}

corrections of post-post-Newtonian magnitude.
(b) Let a test particle move in a bound orbit in the equatorial plane. Use Hamilton-Jacobi theory to show that its orbit is a precessing ellipse [equation (40.17)] with a precession per orbit given by

\begin{matrix} (40.26) & δ {\dot{ϕ}}_{0} = \frac{2 - β + 2 γ}{3} \frac{6 π M_{⊙}}{a (1 - e^{2})} + J_{2} \frac{3 π R_{⊙}^{2}}{a^{2} {(1 - e^{2})}^{2}} \end{matrix}

For the significance of this result, see Box 40.3.

The rest of this chapter is Track 2 . No earlier Track-2 material is needed as preparation for it, but the following will be helpful:
(1) Chapter 6 (accelerated observers);
(2) §17.6 (no prior geometry); and
(3) Chapters 38 and 39 (tests of foundations; other theories; PPN formalism). It is not needed as preparation for any later chapter.

3-body effects in lunar orbit:
(1) theory
(2) prospects for measurement

§40.6. THREE-BODY EFFECTS IN THE LUNAR ORBIT

The relativistic effects discussed thus far all involve the spherical part of the sun's external gravitational field, and thus they can probe only the PPN parameters

β

and

γ

plus the "preferred-frame" parameters

α_{1}, α_{2}

, and

α_{3}

. Attempts to measure other PPN parameters can focus on three-body interactions (discussed here), the dragging of inertial frames by a rotating body (

§ 40.7

), anomalies in the locally measured gravitational constant (

$ 40.8

), and deviations of planetary and lunar orbits from geodesics (

$ 40.9

There is no better place to study three-body interactions than the Earth-moon orbit. The pulls of the Earth, the moon, and the sun all contribute. Perturbations in the motion of Earth and moon about their common center of gravity can be measured with high precision using laser ranging (earth-moon separation measured to

\sim 10 cm

in early 1970 's) and using a radio beacon on the moon's surface (angular position on sky potentially measurable to better than

0^{''} .001

of arc).

Over and above any Newtonian three-body interactions, the Earth and the sun, acting together in a nonlinear manner, should produce relativistic perturbations in the lunar orbit that are barely within the range of this technology. These effects depend on the familiar parameters

γ

(measuring space curvature) and

β

[measuring amount of nonlinear superposition,

{(U_{Earth} + U_{sun})}^{2}

, in

g_{00}]

. In addition, they depend on

β_{2}

, which regulates the extent to which the sun's potential,

U_{sun}

, acting inside the Earth, affects the strength of the Earth's gravitational pull, causing it to vary as the Earth moves nearer and farther from the sun. These effects are expected to depend also on

ζ, Δ_{1}

, and

Δ_{2}

, which regulate the extent to which the Earth's orbital momentum and anisotropies in kinetic energy (caused by the sun) gravitate.

Bromberg (1958), Baierlein (1967), and Krogh and Baierlein (1968) have calculated the three dominant three-body effects in the Earth-moon orbit using general relativity and the Dicke-Brans-Jordan theory. These effects are noncumulative and have amplitudes of

\sim 100 cm, \sim 20 cm

, and

\sim 10 cm

. The

100 - cm

effect [which was originally discovered by de Sitter (1916)] is known to depend only on

γ

. The precise dependence of the other effects on the PPN parameters is not known.

The prospects for measuring these effects in the 1970's are dim; they are masked by peculiarities in the orbit of the moon that have nothing to do with relativity.

§40.7. THE DRAGGING OF INERTIAL FRAMES

The experiments discussed thus far study the motion of electromagnetic waves, spacecraft, planets, and asteroids through the solar system. An entirely different type of experiment measures changes in the orientation of a gyroscope moving in the gravitational field of the Earth. This experiment is particularly important because it can measure directly the "dragging of inertial frames" by the angular momentum of the Earth.

It is useful, before specializing to a rotating Earth, to derive a general expression for the precession of a gyroscope in the post-Newtonian limit. (Track-1 readers, and others who have not studied Chapters 6 and 39, may have difficulty following the derivation. No matter. It is the answer that counts!)

Let

S^{α}

be the spin of the gyroscope (i.e., its angular momentum vector), and let

u^{α}

be its 4 -velocity. The spin is always orthogonal to the 4 -velocity,

S^{α} u_{α} = 0

(see Box 5.6). Assume that any nongravitational forces acting on the gyroscope are applied at its center of mass, so that there is no torque in its proper reference frame. Then the gyroscope will "Fermi-Walker transport" its spin along its world line (see §6.5):

\begin{matrix} (40.27) & \nabla_{u} S = u (a \cdot S), a \equiv \nabla_{u} u = 4-acceleration. \end{matrix}

The objective of the calculation is to write down and analyze this transport equation in the post-Newtonian limit.

The gyroscope moves relative to the PPN coordinate grid with a velocity

v_{j} \equiv

d x^{j} / d t \equiv d x_{j} / d t

. Assume that

v_{j} \leq ϵ

, where

ϵ

is the post-Newtonian expansion parameter (

ϵ^{2} \approx M_{⊙} / R_{⊙}

). As the gyroscope moves, it carries with itself an orthonormal frame

e_{\hat{α}}

, which is related to the PPN coordinate frame by a pure Lorentz boost, plus a renormalization of the lengths of the basis vectors [transformation (39.41)]. The spin is a purely spatial vector

(S^{0} = 0)

in this comoving frame; its length

{(S_{j} S_{j})}^{1 / 2}

remains fixed (conservation of angular momentum); and its direction is regulated by the Fermi-Walker transport law.

The basis vectors

e_{\hat{α}}

of the comoving frame are not Fermi-Walker transported, by contrast with the spin. Rather, they are tied by a pure boost (no rotation!) to the PPN coordinate grid, which in turn is tied to an inertial frame far from the solar system, which in turn one expects to be fixed relative to the "distant stars." Thus, by calculating the precession of the spin relative to the comoving frame,

\begin{matrix} (40.28) & d S_{j} / d τ \equiv ε_{j \hat{k} ı} Ω_{\hat{k}} S_{ı}, \end{matrix}

one is in effect evaluating the spin's angular velocity of precession,

Ω_{j}

, relative to a frame fixed on the sky by the distant stars.

Calculate

d S_{j} / d τ

\begin{matrix} (40.29) & \frac{d S_{\hat{j}}}{d τ} = \nabla_{u} (S \cdot e_{j}) = (\nabla_{u} S) \cdot e_{j} + s \cdot (\nabla_{u} e_{j}) = s \cdot \nabla_{u} e_{\hat{j}} . \end{matrix}

Here use is made of the fact that

\nabla_{u} S

is in the

u

direction [equation (40.27)] and

Gyroscope precession:
(1) general analysis
is thus orthogonal to

e_{\hat{j}}

. The quantity

S \cdot \nabla_{u} e_{\hat{j}}

is readily evaluated in the PPN coordinate frame. In the evaluation, one uses as metric coefficients [equations (39.32)] the expressions

\begin{aligned} g_{00} = - 1 + 2 U + O (ϵ^{4}), g_{j k} = δ_{j k} (1 + 2 γ U) + O (ϵ^{4}), \\ (40.30) & g_{0 j} = - \frac{7}{2} Δ_{1} V_{j} - \frac{1}{2} Δ_{2} W_{j} + (\binom{"preferred-}{frame terms"}) + O (ϵ^{5}); \end{aligned}

one takes as the components of

e_{\hat{j}}

and

S

[obtained via the transformation (39.41)] the expressions

\begin{matrix} e_{j}^{0} = v_{j} + O (ϵ^{3}), e_{j}^{k} = (1 - γ U) δ_{j k} + \frac{1}{2} v_{k} v_{j} + O (ϵ^{4}), \\ (40.31) & S^{0} = v_{j} S_{\hat{j}} + O (ϵ^{3} S_{j}), \\ S^{k} = (1 - γ U) S_{\hat{k}} + \frac{1}{2} v_{k} v_{j} S_{\hat{j}} + O (ϵ^{4} S_{j}); \end{matrix}

and one uses the relation

\begin{matrix} (40.32) & d v_{j} / d τ = a_{j} + U_{, j} + O (ϵ^{2} U_{, j}) \end{matrix}

where

a_{j}

(assumed

≦ U_{, j}

) are the components of the 4 -acceleration. One finds (see exercise 40.6 ) for the precession of the spin the result

d S_{j} / d τ = S \cdot \nabla_{u} e_{j} = S_{k} [v_{[j} a_{k]} + g_{0 [k, j]} - (2 γ + 1) v_{[j} U_{, k]}] .

Rewritten in three-dimensional vector form this result becomes
(2) general PPN formula for precession
(3) specialization: Thomas precession

\begin{matrix} (40.33a) & d S / d τ = Ω \times S \\ (40.33b) & Ω \equiv - \frac{1}{2} v \times a - \frac{1}{2} \nabla \times g + (γ + \frac{1}{2}) v \times \nabla U \\ (40.33c) & g \equiv g_{0 j} e_{\hat{j}} . \end{matrix}

In this final answer it does not matter whether the 3 -vectors entering into

Ω

are evaluated in the coordinate frame or in the comoving orthonormal frame, since

e_{3}

and

\partial / \partial x_{j}

differ only by corrections of order

ϵ^{2}

.
Equations (40.33) describe in complete generality at the post-Newtonian level of approximation the precession of the gyroscope spin

S

relative to the comoving orthonormal frame that is rotationally tied to the distant stars.

For an electron with spin

S

in orbit around a proton, only the first term,

- \frac{1}{2} ν \times a

, is present (no gravity). This term leads to the Thomas precession, which plays an important role in the fine structure of atomic spectra [see, e.g., Ruark and Urey (1930)]. For other ways of deriving the Thomas precession, see exercise 6.9 and

$ 41.4

The Thomas precession comes into play for a gyroscope on the surface of the Earth (

a =

Newtonian acceleration of gravity), but not for a gyroscope in a freely moving satellite.

If one ignores the rotation of the Earth and preferred-frame effects, and puts the PPN coordinate frame at rest relative to the center of the Earth, then

g_{0 j}

vanishes and

Ω

is given by

\begin{aligned} Ω & = ν \times [- \frac{1}{2} a + (γ + \frac{1}{2}) \nabla U] \\ (40.34) & = γ ν \times \nabla U for gyroscope on Earth's surface \\ = (γ + \frac{1}{2}) ν \times \nabla U for gyroscope in orbit. \end{aligned}

The general-relativistic term

(γ + \frac{1}{2}) v \times \nabla U

is caused by the motion of the gyroscope through the Earth's curved, static spacetime geometry. Notice that it depends solely on the same parameter

γ

as is tested by electromagnetic-wave experiments. In order of magnitude, for a gyroscope in a near-Earth, polar orbit,

\begin{matrix} (40.35) & Ω \approx \frac{3}{2} {(\frac{M_{E}}{R_{E}})}^{1 / 2} (\frac{M_{E}}{R_{E}^{2}}) \approx 8 seconds of arc per year. \end{matrix}

The general-relativistic precession

\frac{3}{2} v \times \nabla U

was derived by W. de Sitter (1916) for the "Earth-moon gyroscope" orbiting the sun. Eleven years later L. H. Thomas (1927) derived the special relativistic precession

- \frac{1}{2} v \times a

for application to atomic physics.

The Earth's rotation produces off-diagonal terms,

g_{0 j}

, in the PPN metric (exercise 40.7):

\begin{matrix} (40.36) & g = g_{0 j} e_{j} = - (\frac{7}{4} Δ_{1} + \frac{1}{4} Δ_{2}) \frac{J \times r}{r^{3}} \end{matrix}

Here

J

is the Earth's angular momentum. These off-diagonal terms contribute an amount

\begin{matrix} (40.37) & Ω = - \frac{1}{2} \nabla \times g = (\frac{7}{8} Δ_{1} + \frac{1}{8} Δ_{2}) \frac{1}{r^{3}} [- J + \frac{3 (J \cdot r) r}{r^{2}}] \end{matrix}

to the precession of the gyroscope. Notice that this contribution, unlike the others, is independent of the linear velocity of the gyroscope. One can think of it in the following way.

The gyroscope is rotationally at rest relative to the inertial frames in its neighborhood. It and the local inertial frames rotate relative to the distant galaxies with the angular velocity

Ω

because the Earth's rotation "drags" the local inertial frames along with it. Notice that near the north and south poles the local inertial frames rotate in the same direction as the Earth does (

Ω

parallel to

J

), but near the equator they rotate in the opposite direction (

Ω

antiparallel to

J

; compare

Ω

with the magnetic field of the Earth!). Although this might seem paradoxical at first, an analogy devised by Schiff makes it seem more reasonable.* Consider a rotating, solid sphere immersed in a viscous fluid. As it rotates, the sphere will drag the fluid along with it. At various points in the fluid, set down little rods, and watch how the fluid

(4) specialization: precessions due to acceleration and Earth's Newtonian potential
(5) specialization: precession due to Earth's rotation
rotates them as it flows past. Near the poles the fluid will clearly rotate the rods in the same direction as the star rotates. But near the equator, because the fluid is dragged more rapidly at small radii than at large, the end of a rod closest to the sphere is dragged by the fluid more rapidly than the far end of the rod. Consequently, the rod rotates in the direction opposite to the rotation of the sphere.

In order of magnitude, the precessional angular velocity caused by the Earth's rotation is

\begin{matrix} (') & Ω \sim J_{E} / R_{E}^{3} \sim 0.1 seconds of arc per year. \end{matrix}

(6) prospects for measuring precession

Both this precession, and the larger one [equation (40.35)] due to motion through the Earth's static field, may be detectable in the 1970's. Equipment aimed at detecting them via a satellite experiment is now (1973) under construction at Stanford University; see Everitt, Fairbank, and Hamilton (1970); also O'Connell (1972).*

The gyroscope precession produced by motion of the Earth relative to the preferred frame (if any) is too small to be of much interest.

EXERCISES

Exercise 40.6. PRECESSIONAL ANGULAR VELOCITY

Derive equations (40.33) for the precession of a gyroscope in the post-Newtonian limit. Base the derivation on equations (40.29)-(40.32).

Exercise 40.7. OFF-DIAGONAL TERMS IN METRIC ABOUT THE EARTH

Idealize the Earth as an isolated, rigidly rotating sphere with angular momentum

J

. Use equations (

39.34 b, c

) and (39.27) to show that (in three-dimensional vector notation)

\begin{matrix} (40.38) & V \equiv V_{j} e_{j} = W \equiv W_{j} e_{j} = \frac{1}{2} J \times r / r^{3} \end{matrix}

outside the Earth, in the Earth's PPN rest frame. From this, infer equation (40.36).

Exercise 40.8. SPIN-CURVATURE COUPLING

Consider a spinning body (e.g., the Earth or a gyroscope or an electron) moving through curved spacetime. Tidal gravitational forces produced by the curvature of spacetime act on the elementary pieces of the spinning body. These forces should depend not only on the positions of the pieces relative to the center of the object, but also on their relative velocities. Moreover, the spin of the body,

S \equiv \int (ρ r \times v) d (volume) in comoving orthonormal frame

is a measure of the relative positions and velocities of its pieces. Therefore one expects the spin to couple to the tidal gravitational forces-i.e., to the curvature of spacetime-producing
deviations from geodesic motion. Careful solution of the PPN equations of Chapter 39 for general relativity reveals [Papapetrou (1951), Pirani (1956)] that such coupling occurs and causes a deviation of the worldline from the course that it would otherwise take; thus,

\begin{matrix} (40.39) & m \frac{D u^{α}}{d τ} = - S_{μ} u_{ν} \frac{D^{2} u_{β}}{d τ^{2}} ϵ^{α μ ν β} + \frac{1}{2} (ϵ^{λ μ ρ τ} R^{α ν}_{λ μ}) u_{ν} S_{ρ} u_{τ} \end{matrix}

Evaluate, in order of magnitude, the effects of the supplementary term on planetary orbits in the solar system.
[Answer: They are much too small to be detected. However, there are two other material places to look for the effect: (1) when a rapidly spinning neutron star, or a black hole endowed with substantial angular momentum enters the powerful tidal field of another neutron stur or black hole; and (2) when an individual electron, or the totality of electrons in the "Dirac sea of negative energy states," enter a still more powerful tidal field (late phase of gravitational collapse). Such a tidal field, or curvature, pulls oppositely on electrons with the two opposite directions of spin [Pirani (1956); DeWitt (1962), p. 338; Schwinger (1963a,b)] just as an electric field pulls oppositely on electrons with the two opposite signs of charge ["vacuum polarization"; see especially Heisenberg and Euler (1936)]. In principle, the tidal field pulling on the spin of an electron need not be due to "background" spacetime curvature; it might be due to a nearby massive spinning object, such as a "live" black hole (chapter 33) ["gravitational spin-spin coupling"; O’Connell (1972)].

§40.8. IS THE GRAVITATIONAL CONSTANT CONSTANT?

The title and subject of this section are likely to arouse confusion. Throughout this book one has used geometrized units, in which

G = c = 1

. Therefore, one has locked oneself into a viewpoint that forbids asking whether the gravitational constant changes from event to event.

False! One can perfectly well ask the question in the context of

G = c = 1

, if one makes clear what is meant by the question.
In

§ § 1.5

and

1.6, c

was defined to be a certain conversion factor between centimeters and seconds; and

G / c^{2}

was defined to be a certain conversion factor between grams and centimeters. These definitions by fiat do not guarantee, however, that a Cavendish experiment* to measure the attraction between two bodies will yield

Force = - G m_{1} m_{2} / r^{2} = - m_{1} m_{2} / r^{2}

If general relativity correctly describes classical gravity, and if the values of the conversion factors

G

and

c

have been chosen precisely right, then any Cavendish experiment, anywhere in the universe, will yield "Force

= - m_{1} m_{2} / r^{2}

". But if the

"Cavendish gravitational constant,"

G_{C}

, defined

Changes of

G_{C}

with time

Spatial variations in

G_{C}

Dicke-Brans-Jordan theory, or almost any other metric theory gives the correct description of gravity, the force in the Cavendish experiment will depend on where and when the experiment is performed, as well as on

m_{1}, m_{2}

, and

r

. To discuss Cavendish experiments as tests of gravitation theory, then, one must introduce a new proportionality factor

\begin{matrix} (40.40) & G_{C} \equiv G_{Cavendish} \equiv ("Cavendish gravitational constant"), \end{matrix}

which enters into the Newtonian force law

\begin{matrix} (40.41) & Force = - G_{C} m_{1} m_{2} / r^{2} \end{matrix}

This Cavendish constant will be unity in general relativity, but in most other metric theories it will vary from event to event in spacetime.
In some theories, such as Dicke-Brans-Jordan, the Cavendish constant is determined by the distribution of matter in the universe. As a result, the expansion of the universe changes its value:

\frac{1}{G_{C}} \frac{d G_{C}}{d t} \sim - (\frac{0.1 to 1}{age of universe}) \sim \frac{- 1}{10^{10} or 10^{11} years}

[see, e.g. Brans and Dicke (1961)]. A variety of observations place limits on such time variations. Big time changes in

G_{C}

during the last 4.6 billion years would have produced marked effects on the Earth, the sun, and the entire solar system. The expected geophysical effects have been summarized and compared with observations by Dicke and Peebles (1965). It is hard to draw firm conclusions because of the complexity of the geophysics involved, but a fairly certain limit is

\begin{matrix} (40.42a) & (1 / G_{C}) (d G_{C} / d t) ≦ 1 / 10^{10} years (geophysical). \end{matrix}

Eventually, high-precision measurements of the orbital motions of planets will yield a better limit. For the present, planetary observations show

\begin{matrix} (40.42b) & (1 / G_{C}) (d G_{O} / d t) ≦ 4 / 10^{10} years (planetary orbits) \end{matrix}

[Shapiro, Smith, et al.(1971)]. These limits are tight enough to begin to be interesting, but not yet tight enough to disprove any otherwise viable theories of gravity.
If

G_{C}

is determined by the distribution of matter in the universe, then it should depend on where in the universe one is, as well as when. In particular, as one moves from point to point in the solar system, closer to the Sun and then farther away, one should see

G_{C}

change. Indeed this is the case in most metric theories of gravity, though not in general relativity. Analyses of Cavendish experiments using the PPN formalism reveal spatial variation in

G_{C}

given by

\begin{matrix} (40.43) & Δ G_{C} = - 2 G_{C} (β + γ - β_{2} - 1) U \end{matrix}

[Nordtvedt (1970, 1971a); Will (1971b)].
The amplitude of these variations along the Earth's elliptical orbit is

Δ G_{C} / G_{C} \sim

10^{- 10}

, if

β + γ - β_{2} - 1 \sim 1

. This is far too small to measure directly in the

1970's. Despite great ingenuity and effort, the most accurate experiments measuring the value of

G_{C}

have precisions in 1972 no better than 1 part in

10^{4}

[see Beams (1971)]. Experiments to search for yearly variations in

G_{C}

on Earth without measuring the actual value ("null-type experiments") can surely be performed with better precision than 1 in

10^{4}

-but not with precisions approaching 1 in

10^{10}

. On the other hand, indirect consequences of a spatial variation of

G_{C}

in the solar system are almost certainly measurable (see

§ 40.9

below).

In Ni's theory of gravity (Box 39.1), and other two-tensor or vector-tensor theories like it, where the prior geometry picks out a preferred "universal rest frame," the Cavendish constant

G_{C}

can depend on velocity relative to the preferred frame. For Cavendish experiments with two equal masses separated by distances large compared to their sizes,

G_{C}

varies as

\begin{matrix} (40.44) & Δ G_{C} = G_{C} [\frac{1}{2} (α_{2} + α_{3} - α_{1}) v^{2} - \frac{1}{2} α_{2} (v \cdot n)^{2}] \end{matrix}

[Will (1971b)]. Here

v

is the velocity of the Cavendish apparatus relative to the preferred frame, and

n

is the unit vector between the two masses. For experiments where one body is a massive sphere (e.g., the Earth), and the other is a small object on the sphere's surface,

G_{C}

varies as

\begin{aligned} Δ G_{C} / G_{C} = & \frac{1}{2} [(α_{3} - α_{1}) + α_{2} (1 - I / M R^{2})] v^{2} \\ (') & - \frac{1}{2} α_{2} (1 - 3 I / M R^{2}) (v \cdot n)^{2} \end{aligned}

[Nordtvedt and Will (1972)]. Here

M

and

R

are the mass and radius of the sphere, and

I = \int (ρ r^{2}) 4 π r^{2} d r

is the trace of the second moment of its mass distribution. Consequences of these effects for planetary orbits have not yet been spelled out, but consequences for Earthbound experiments have.

Think of a Cavendish experiment in which one mass is the Earth, and the other is a gravimeter on the Earth's surface. The gravimeter gives a reading for the "local acceleration of gravity,"

\begin{matrix} (40.45) & g = G_{C} m_{Earth} / r_{Earth}^{2} \end{matrix}

As the Earth turns, so the unit vector

n

between its center and the gravimeter rotates,

G_{C}

and hence

g

will fluctuate with a period of 12 sidereal hours and an amplitude

(Δ g / g)_{amplitude} = \frac{1}{4} α_{2} v^{2} \cos^{2} θ_{m}

Here

θ_{m}

is the minimum, as the Earth rotates, of the angle between

v

(constant vector) and

n

(rotating vector). (Note: we have used the value

I / M R^{2} ≃ 0.5

for the Earth.) These fluctuations will produce tides in the Earth of the same type as are

Dependence of

G_{C}

on velocity

Anomalies in Earth tides due to anisotropies in

G_{C}

(1) experimental value of

α_{2}

(2) experimental disproof of Whitehead theory

Anomalies in Earth rotation rate due to dependence of

G_{C}

on velocity
produced by the moon and sun. As of 1972, gravimeter measurements near the Earth's equator show no sign of any anomalous 12 -sidereal-hour effects down to an amplitude of

\sim 10^{- 9}

[Will (1971b)]. Consequently,

\begin{matrix} (40.46a) & {| α_{2} |}^{1 / 2} v \cos θ_{m} = {| Δ_{2} + ζ - 1 |}^{1 / 2} v \cos θ_{m} ≦ 6 \times 10^{- 5} \sim 20 km / \sec \end{matrix}

Using a rough estimate of

v \sim 200 km / \sec

for the Earth's velocity relative to the universal rest frame, and

θ_{m} \leq 60^{\circ}

for the angle between

v

and the Earth's equatorial plane, one obtains the rough limit

\begin{matrix} (40.46b) & | α_{2} | = | Δ_{2} + ζ - 1 | ≦ 0.03 \end{matrix}

[This limit does not affect the three theories in Box 39.1; of them, only Ni's theory has prior geometry and a universal rest frame; and it predicts isotropic effects in

Δ G_{C} / G_{C}

[equation (40.44)], but no anisotropic effects. However, other theories with universal rest frames-e.g. Papapetrou's (1954a,b,c) theory-are ruled out by this limit; see Ni (1972), Nordtvedt and Will (1972).]

Whitehead's theory of gravity (which is a two-tensor theory with a rather different type of prior geometry from Ni's) predicts that the galaxy should produce velocityindependent anisotropies in

G_{C}

. These, in turn, would produce Earth tides with periods of 12 sidereal hours and amplitudes of

Δ g / g \sim 2 \times 10^{- 7} \sim 100 \times (\binom{experimental limit on}{such amplitudes})

[Will (1971b)]. The absence of such tides proves Whitehead's theory to be incor-rect-a feat of disproof beyond the power of all redshift, light-deflection, time-delay, and perihelion-shift measurements. (For all these "standard experiments," the predictions of Whitehead and Einstein are identical!)

Equation

({40.44}^{'})

predicts a periodic annual variation of the Cavendish constant on Earth, as the Earth moves around the sun:

\begin{matrix} v = (\binom{velocity of Earth}{relative to sun}) + (\binom{velocity of sun relative}{to preferred frame}) \equiv v_{E} + w; \\ (40.47) & {(Δ G_{C} / G_{C})}_{\begin{matrix} averaged over \\ Surface of Earth \end{matrix}} = \frac{1}{2} (\frac{2}{3} α_{2} + α_{3} - α_{1}) (w^{2} + v_{E}^{2} + 2 w \cdot v_{E}) \end{matrix}

This annual variation, assuming all PPN parameters are of order unity, is 1,000 times larger than the one produced by the Earth's motion in and out through the sun's gravitational potential [equation (40.43)]. In response to this changing Cavendish constant, the Earth's self-gravitational pull should change, and the Earth should "breathe" inward (greater pull) and outward (relaxed pull). The resulting annual variations in the Earth's moment of inertia should produce annual changes in its rotation rate

ω

(changes in "length of day" as measure by atomic clocks):

δ ω / ω \sim 0.1 (\frac{2}{3} α_{2} + α_{3} - α_{1}) w \cdot v_{E}

[Nordtvedt and Will (1972)]. Comparison with the measured annual variations of rotation rate (all of which geophysicists attribute to seasonal changes in the Earth's atmosphere) yields the following limit

\begin{matrix} (40.48) & | \frac{2}{3} α_{2} + α_{3} - α_{1} | =≦ 0.2 \end{matrix}

[See Nordtvedt and Will (1972)]. This limit rules out several preferred-frame theories of gravity, including that of Ni (Boxes 39.1 and 39.2).
The experimental results

(40.21), (40.46)

, and (40.48), when combined, place the following very rough limits on any theory that possesses a Universal rest frame:

\begin{aligned} | α_{1} | = | 7 Δ_{1} + Δ_{2} - 4 γ - 4 | ≦ 0.2, \\ (40.49) & | α_{2} | = | Δ_{2} + ζ - 1 | ≦ 0.03, \\ | α_{3} | = | 4 β_{1} - 2 γ - 2 - ζ | ≦ 2 \times 10^{- 5} . \end{aligned}

These limits completely disprove all theories with preferred frames that have been examined to date except one devised by Will and Nordtvedt [see Ni (1972); Nordtvedt and Will (1972)].

In some theories of gravity, the result of a Cavendish experiment depends on the chemical composition and internal structure of the test bodies (exercises 40.9 and 40.10). Kruezer (1968) has performed the most accurate search for such effects to date. He finds that

G_{C}

is the same for fluorine and bromine to a precision of

\begin{matrix} (40.50) & | \frac{G_{C} (bromine) - G_{C} (fluorine)}{G_{C}} | ≦ 5 \times 10^{- 5} . \end{matrix}

Exercise 40.9. CAVENDISH CONSTANT FOR IDEALIZED SUN

Idealize the sun as a static sphere of perfect fluid at rest at the origin of the PPN coordinates. Then its external gravitational field has the form (40.3), with

M_{⊙}

given by (40.4). Consequently, a test body of mass

m

, located far away at radius

r

, is accelerated by a gravitational force

\begin{matrix} (40.51a) & Force = - m M_{⊙} / r^{2} . \end{matrix}

(a) Calculate the mass of the sun,

M

, in the sense of the amount of energy required to construct it by adding one spherical shell of matter on top of another, working from the inside outward. [Answer:

\begin{aligned} M & = \underset{rest mass + internal energy}{\underset{⏟}{\int_{0}^{R_{⊙}} ρ_{0} (1 + Π + 3 γ U) 4 π r^{2} d r}} \underset{gravitational potential energy}{\underset{⏟}{- \frac{1}{2} \int_{0}^{R_{⊙}} ρ_{0} U 4 π r^{2} d r}} \\ = \int_{0}^{R_{⊙}} ρ_{0} [1 + Π + (3 γ - \frac{1}{2}) U] 4 π r^{2} d r .] \end{aligned}

(b) Use the virial theorem [equation (39.21b)] to rewrite equation (40.4) in the form

\begin{matrix} (40.51c) & M_{⊙} = \int_{0}^{R_{⊙}} ρ_{0} [1 + β_{3} Π + (2 β_{2} + \frac{1}{2} β_{4}) U] 4 π r^{2} d r . \end{matrix}

EXERCISES

\begin{matrix} (40.51d) & Force = - G_{C} m M / r^{2} \end{matrix}

of the Cavendish constant for

r

far outside the sun, to obtain

\begin{aligned} (40.52) & G_{C} & = \frac{(\binom{mass of sun as defined by its effect in}{bending world line of a faraway test particle})}{(\begin{matrix} mass-energy as defined by applying law of \\ conservation of energy to the steps in the \\ construction of the sun \end{matrix}} \\ = 1 + \int (ρ_{0} / M_{0}) [(β_{3} - 1) Π + \frac{1}{2} (4 β_{2} + β_{4} - 6 γ + 1) U] 4 π r^{2} d r \end{aligned}

Unless

β_{3} = 1

, and

4 β_{2} + β_{4} - 6 γ + 1 = 0

(as they are, of course, in Einstein's theory),

G_{C}

will depend on the sun's internal structure! Specialize equation (40.52) to "conservative theories of gravity (Box 39.5), and explain why the result is what one would expect from equation (40.43).

Exercise 40.10. CAVENDISH CONSTANT FOR ANY BODY

Extend the analysis of exercise 40.9 to a source that is arbitrarily stressed and has arbitrarv shape and internal velocities (subject to the constraints

v^{2} ≪ 1, | t_{j k} | / ρ_{0} ≪ 1, U ≪ 1, Π ≪ 1

, of the post-Newtonian approximation). Assume that the body is at rest relative to the universal rest frame. Show that

G_{C}

depends on the internal structure of the source unless

\begin{matrix} (40.53) & 2 β_{1} - β_{4} = 1, 4 β_{2} + β_{4} - 6 γ = - 1, β_{3} = 1, ζ = 0, η = 0 \end{matrix}

Of course, these PPN constraints are all satisfied by Einstein's theory.

The sense in which general relativity predicts geodesic motion for planets and sun

§40.9. DO PLANETS AND THE SUN MOVE ON GEODESICS?

Crucial to solar-system experiments is the question of whether the sun and the planets move on geodesics of spacetime. This question is complicated by the contributions to the spacetime curvature made by the moving body itself.

To elucidate the question-and to obtain an answer in the framework of general relativity-consider an "Einstein elevator" type of argument. The astronomical object under consideration has an outer boundary, and each point on this boundary describes a world line. These world lines define a world tube. Some distance outside of this world tube construct a "buffer zone" as in §20.6. Tailor its inner and outer dimensions, according to the mass and moments of the object and the curvature of the enveloping space ("strength of the tide-producing force of the external gravitational field"), in such a way that the departure

ϵ

(cf. §20.6) of the metric from flatness in this buffer zone takes on values equal at most to twice the extremal achievable value

ϵ_{extrem}

(a minimum with respect to variations in

r

, a maximum
with respect to variations in direction; in other words, a minimax). Then, apart from errors of order

ϵ_{extrem}

, the object can be regarded as moving in an asymptotically flat space. The law of conservation of total 4-momentum applies. It assures one that the object moves in a (locally) straight line with uniform velocity. Consider, next, a "background geometry" that agrees just outside the buffer zone with the actual geometry to accuracy

ϵ_{extrem}

or better, but that inside is a source-free solution of Einstein's field equation. Then, to an accuracy governed by the magnitude of

ϵ_{extrem}

, the locally straight line along which the astronomical object moves will be a geodesic of this background geometry.

Insofar as one can give any well-defined meaning to the departure of the actual motion from this geodesic (a task complicated by the fact that the background geometry does not actually exist!), one can calculate this departure by making use of the PPN formalism or some other approximation scheme [see, e.g., Taub (1965)]. This deviation springs ordinarily in substantial measure, and sometimes almost wholly, from a coupling between the Riemann curvature tensor of the external field and the multipole moments of the astronomical object (angular momentum associated with rotation; quadrupole and higher moments associated with deformation; see, e.g., exercises 40.8 and 16.4). This coupling is important for the Earth-moon system, but one need not use relativity to calculate it; Newtonian theory does the job to far greater accuracy than needed-or would, if one understood the interiors of the Earth and the moon well enough! For the planets and sun, the effect is negligible. (Exercise: use Newtonian theory to prove so!).

Thus, in general relativity as applied to the solar system, one can approximate the orbit of the sun, the Earth-moon mass center, and each other planet, as a geodesic of that "background spacetime geometry" which would exist if its own curvature effects were absent. This is the approach used to analyze the perihelion shift for planets in

§ 40.5

in the context of general relativity, and to derive in exercise 39.15 the post-Newtonian "many-body equations of motion."

In most other metric theories of gravity, including the Dicke-Brans-Jordan theory, there are substantial departures from geodesic motion. The "Einstein elevator" argument fails in these theories because spacetime is endowed not only with a metric, but also with a long-range field that couples indirectly (cf.

§ § 38.7

and 39.2 ) to massive, gravitating bodies.

This phenomenon is best understood in terms of Dicke's argument about the influence of spatial variations of the fundamental constants on experiments of the Eötvös-Dicke type (see §38.6). In a theory where the Cavendish gravitational constant

G_{C}

depends on position (as it does not and cannot in general relativity), a body with significant self-gravitational energy

E_{grav}

must fall, in a perfectly uniform external Newtonian gravitational field, with an anomalous acceleration:

\begin{aligned} (40.54) & (\binom{acceleration of}{massive body}) - (\binom{acceleration of}{test body}) & = \frac{1}{M} (\frac{\partial E_{grav}}{\partial G_{C}}) \nabla G_{C} \\ = \frac{E_{grav}}{M G_{C}} \nabla G_{C} \end{aligned}

Deviations from geodesic motion:
(1) due to curvature coupling
(2) due to spatial dependence of gravitational constant (Nordtvedt effect)
[see equation (38.15)]. In Dicke-Brans-Jordan theory,

G_{C}

is essentially the reciprocal of the scalar field; and it contains a small part that is proportional to the Newtonian potential,

U

[equation (40.43) with the appropriate values of the parameters from Box 39.2]. As a result, the sun falls with an acceleration smaller by one part in

10^{6}

than the acceleration of a test body; Jupiter falls with an acceleration one part in

10^{9}

smaller; and the Earth, one part in

10^{10}

smaller. Translated into relativistic language: the scalar field, by influencing the gravitational self-energy of a massive body, produces deviations from geodesic motion.

One can use the full PPN formalism of Chapter 29 to calculate the motion of massive bodies in any metric theory of gravity. Nordtvedt (1968b) and Will (1971a) have done this. They find that a massive body at rest in a uniform external field experiences a (Newtonian-type) PPN coordinate acceleration given by

\frac{d^{2} x_{j}}{d t^{2}} = E_{j k} \frac{\partial U}{\partial x_{k}}

where

E_{j k}

is a quantity depending on the body's structure:

\begin{matrix} E_{j k} = δ_{j k} {1 - (7 Δ_{1} - 3 γ - 4 β) \frac{E_{grav}}{m}} - (2 β + 2 β_{2} - 3 γ + Δ_{2} - 2) \frac{Ω_{j k}}{m} \\ (40.55) & Ω_{j k} = - \frac{1}{2} \int \frac{ρ_{o} ρ_{o}^{'} (x_{j} - x_{j}^{'}) (x_{k} - x_{k}^{'})}{{| x - x^{'} |}^{3}} d^{3} x d^{3} x^{'}, E_{grav} = \sum Ω_{j j} \end{matrix}

Here

m

is the body's total mass-energy,

Ω_{j k}

is the "Chandrasekhar potential-energy tensor," and

E_{grav}

is the body's self-gravitational energy. [Note: Dicke's method of calculating the anomalous acceleration (40.54) breaks down in theories that are not "conservative" (Box 39.5).]
In general relativity, the combinations of PPN coefficients appearing in

E_{j k}

vanish; so

E_{j k} = δ_{j k}

, and the body falls with the usual acceleration-i.e., it moves along a geodesic. But in most other theories of gravity

E_{j k} \neq δ_{j k}

; the body does not move on a geodesic; and its acceleration may even be in a different direction than the gradient of the Newtonian potential!

This predicted departure from geodesic motion is called the "Nordtvedt effect." The possibility of such an effect was first noticed in passing by Dicke (1964c), but was discovered independently and explored in great detail by Nordtvedt (1968a,b). The Nordtvedt effect in a theory other than general relativity produces a number of phenomena in the solar system that are potentially observable. [See Nordtvedt (1971b) for an enumeration and references.] The effect most suitable for a test is a "polarization" of the Earth-moon orbit due to the fact that the moon should fall toward the sun with a greater acceleration than does the Earth. This "polarization" results in an eccentricity in the orbit that points always toward the Sun and has the amplitude

Nordtvedt effect in Earth-moon orbit

\begin{aligned} δ r & = 840 [3 γ + 4 β - 7 Δ_{1} - \frac{1}{3} (2 β + 2 β_{2} - 3 γ + Δ_{2} - 2)] cm \\ = 67 meters & in Ni's theory (Boxes 39.1 and 39.2) \\ = \frac{8.4}{2 + ω} meters & in Dicke-Brans-Jordan theory (Boxes 39.1 and 39.2) \\ = 0 & in Einstein's theory. \end{aligned}

Box 40.4 CATALOG OF EXPERIMENTS

Type of experiment
I. Tests of foundations of general relativity
II. Post-Newtonian
("solar-system") experiments
III. Cosmological observations
IV. Gravitational-Wave experiments

Description of experiment

Tests of uniqueness of free fall (Eötvös-Dicke-Braginsky experiments
Tests for existence of metric (time dilation of particle decays; role of Lorentz group in particle kinematics; etc.)
Searches for new, direct-coupling, long-range fields (HughesDrever experiment; ether-drift experiments)
Gravitational redshift experiments
Constancy, in space and time, of the nongravitational physical constants
Deflection of light and radio waves by Sun
Relativistic delay in round-trip travel time for radar beams passing near Sun
Perihelion shifts and periodic perturbations in planetary orbits
Three-body effects in the Lunar orbit
Precession of gyroscopes ("geodetic precession" and precession due to dragging of inertial frames by Earth's rotation)
Spatial variation of the Cavendish gravitational constant in the solar system
Dependence of the Cavendish gravitational constant on the chemical composition of the gravitating body
Earth tides with sidereal periods
Annual variations in Earth rotation rate
Periodicities in Earth-Moon separation due to breakdown of geodesic motion
Change of Cavendish gravitational constant with time in solar system
Large-scale features of universe (expansion, isotropy, homogeneity; existence and properties of cosmic microwave radiation; . . .)
Agreement of various measures of age of universe (age from expansion; ages of oldest stars; age of solar system)

Existence of waves; propagation speed; polarization properties; ...

Where discussed
§38.3; Figure 1.6; Box 1.1
838.4
§38.7; Figure 38.3
§38.5; Figures 38.1 and

38.2; 887.2

, 7.3 , and 7.4
§38.6
§40.3; Box 40.1
§40.4; Box 40.2
§40.5; Box 40.3

$ 40.6

§40.7

$ 840.8

and 40.9
§40.8
$40.8
§40.8

$ 40.9

§40.8
Chapters 27-30; especially Chapter 29

§29.7

Chapters 35-37; especially Chapter 37

Figure 40.4. (facing page)

Measuring the separation between earth and moon by determining the time-delay (about 2.5 sec ) between the emission of light from a laser on the earth and the return of this light to the earth. A key element in the program is a corner reflector, the first of which was landed on the moon July 20, 1969, by the Apollo 11 flight crew. In November 1971, there were three such reflectors on the moon: two American, and one French-built and Soviet-landed. A pulsed ruby laser projects a beam out of the 107 -inch reflecting telescope of the McDonald Observatory of the University of Texas, on Mount Locke, 119 miles east of El Paso. This beam makes a spot of light on the moon's surface about 3.2 km in diameter. Laser light is bounced straight back to the earth by the "laser ranging retroreflectors"

({LR}^{3})

. Each consists of an aluminum panel of 46 cm by 46 cm with 100 fused silica corner cubes each 3.8 cm in diameter. The first reflector ever set up appears in the first inset, near the lunar landing module. It is tilted with respect to the landscape of the moon. The photograph was made shortly before astronauts Neil A. Armstrong and Edwin E. Aldrin, Jr., took off for the earth. The second inset is a photograph made by D. G. Currie of the field of view in the guiding eyepiece of the McDonald 107 -inch telescope in an interval when the laser was not firing at the Apollo 11 site. One guides the telescope to Tranquility Base (small circle) by aligning fiducial marks on more visible moonscape features. In November 1971, the

{LR}^{3}

experiment and continuing time-of-flight measurements were the responsibility of the National Aeronautics and Space Administration and a Lunar Retroreflecting Ranging Team of representatives from several centers of research. One of the members of this team, Carroll Alley, of the University of Maryland, is hereby thanked for his kindness in providing the photographs used in this montage. Thanks to this NASA work, the distance between the laser source on the earth and the reflectors on the moon is known with an accuracy now better than half a meter. The astronauts left behind on the moon not only

{LR}^{3}

and a seismometer and other equipment, but also a plaque: "We came in peace for all mankind."

By the mid 1970's, lunar laser-ranging data will probably be able to determine the amplitudes of this polarization to a precision of one meter or better [see Bender et al. (1971); also Figure 40.4].

§40.10. SUMMARY OF EXPERIMENTAL TESTS OF GENERAL RELATIVITY

No longer is general relativity "a theorist's Paradise, but an experimentalist's Hell." It is now a Paradise for all-as one can see quickly by perusing the catalog of experiments given in Box 40.4 on page 1129. Moreover, general relativity has emerged from each of its tests unscathed-a remarkable 1973 tribute to the 1915 genius of Albert Einstein.

FRONTIERS

Wherein the reader-who, during a life of continued variety for forty chapters (besides the Preface), was eight chapters a mathematician, four times enticed (once by an old friend), four chapters a cosmologist, and four chapters a transported astrophysicist in the land of black holes, and who at last inherited a wealth of experiments, lived honest, and became a

True Believer-now ventures forth in search of new
frontiers to conquer.

синоттия 41

SPINORS

§41.1. REFLECTIONS, ROTATIONS, AND THE COMBINATION OF ROTATIONS

Spinors and their applications in relativity grew out of the analysis of "rotations," first in space, then in spacetime. Take a cube (Figure 41.1). Rotate it about one axis through

90^{\circ}

. Then pick another axis at right angles to the first. About it rotate the cube again through

90^{\circ}

. In this way the cube is carried from the orientation marked "Initial" to that marked "Final." How can one make this net transformation in a single step, with a single rotation? In other words, what is the law for the combination of rotations?

Were rotations described by vectors, then one could apply the law of combination of vectors. The resultant of two vectors of the same magnitude

(90^{\circ})

separated by a right angle, is a single vector that (1) lies in the same plane and (2) has the magnitude

2^{1 / 2} \times 90^{\circ} = {127.28}^{\circ}

. Both predictions are wrong. To turn the cube from initial to final orientation in a single turn, (1) take an axis running from the center through the vertex

A

and (2) rotate through

120^{\circ}

What computational algorithm can ever reproduce a law of combination of rotations apparently so strange? On the evening of October 16, 1843, William Rowan Hamilton was walking with his wife along the Royal Canal in Dublin when the answer leaped to his mind, the fruit of years of reflection. With his knife he then and there carved on a stone on Brougham Bridge the formulas*

i^{2} = j^{2} = k^{2} = i j k = - 1,

This chapter is entirely Track 2. No earlier Track-2 material is needed as preparation for it, nor is it needed as preparation for any later chapter.

The problem of combining rotations

Figure 41.1.
Rotation about the vertical axis through

90^{\circ}

, followed by rotation about the horizontal axis through

90^{\circ}

, gives a net change in orientation that can be achieved by a single rotation through

120^{\circ}

about an axis emergent from the center through the corner

A

.
which in today's notation,

\begin{matrix} (41.1) & σ_{x} = ‖ \begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array} ‖ = i i, σ_{y} = ‖ \begin{array}{rr} 0 & - i \\ i & 0 \end{array} ‖ = i j, σ_{z} = ‖ \begin{array}{rr} 1 & 0 \\ 0 & - 1 \end{array} ‖ = i k, \end{matrix}

take the form

\begin{aligned} (41.2) & σ_{x}^{2} & = σ_{y}^{2} = σ_{z}^{2} = 1 \\ σ_{x} σ_{y} & = - σ_{y} σ_{x} = i σ_{z} (and cyclic permutations). \end{aligned}

To any rotation is associated a quantity (Hamilton's "quaternion;" today's "spin matrix" or "spinor transformation" or "rotation operator")

\begin{matrix} (41.3) & R = \cos (θ / 2) - i \sin (θ / 2) (σ_{x} \cos α + σ_{y} \cos β + σ_{z} \cos γ), \end{matrix}

where

θ

is the angle of rotation and

α, β, γ

are the angles between the axis of rotation and the coordinate axes. A rotation described by

R_{1}

followed by a rotation described by

R_{2}

gives a net change in orientation described by the single rotation

\begin{matrix} (41.4) & R_{3} = R_{2} R_{1} . \end{matrix}

This is Hamilton's formula for the combination of two rotations (steps toward it by Euler in 1776; obtained by Gauss in 1819 but never published by him).
In the example in Figure 41.1,

\begin{aligned} R_{1} (rotation by θ = 90^{\circ} about z -axis) = (1 - i σ_{z}) / 2^{1 / 2}, \\ R_{2} (rotation by θ = 90^{\circ} about x -axis) = (1 - i σ_{x}) / 2^{1 / 2}, \end{aligned}

and the product of the two is

\begin{aligned} R_{2} R_{1} & = (1 - i σ_{x} + i σ_{y} - i σ_{z}) / 2 \\ = \cos 60^{\circ} - i \sin 60^{\circ} (σ_{x} / 3^{1 / 2} - σ_{y} / 3^{1 / 2} + σ_{z} / 3^{1 / 2}) . \end{aligned}

Figure 41.2.
Reflection in the plane

M P Q

carries

A

B

. Reflection in the plane

N P Q

carries

B

C

. The combination of the two reflections in the two planes separated by the angle

θ / 2

produces the same end result (transformation from

A

C

) as rotation through the angle

θ

about the line

P Q

According to Hamilton's rule (41.3), this result implies a net rotation through

120^{\circ}

about a line that makes equal angles with the

x

-axis, the

y

-axis, and the

z

-axis, in conformity with what one already saw in Figure 41.1 (axis of rotation running from center of cube through the corner

A

What one has just done in the special example one can do in the general case: obtain the parameters

θ_{3}, α_{3}, β_{3}, γ_{3}

of the net rotation (four unknowns!) by identifying the four coefficients of the four Hamilton units

1, - i σ_{x}, - i σ_{y}, - i σ_{z}

on both sides of the equation

R_{3} = R_{2} R_{1}

. In this way one arrives at the four prequaternion formulas of Olinde Rodrigues (1840) for the combination of the two rotations.

Why do half-angles put in an appearance? And what is behind the law of combination of rotations? The answer to both questions is the same: a rotation through the angle

θ

about a given axis may be visualized as the consequence of successive reflections in two planes that meet along that axis at the angle

θ / 2

(Figure 41.2). Two rotations therefore imply four reflections. However, it can be arranged that reflections no. 2 and no. 3 take place in the same plane, the plane that includes the two axes of rotation. Then reflection no. 3 exactly undoes reflection no. 2. By now there remain only reflections no. 1 and no. 4 , which together constitute one rotation: the net rotation that was desired (Figures 41.3 and 41.4).

The rotation

\begin{matrix} (41.3) & R = \cos (θ / 2) - i \sin (θ / 2) (σ_{x} \cos α + σ_{y} \cos β + σ_{z} \cos γ) \end{matrix}

is undone by the inverse rotation

\begin{matrix} (') & R^{- 1} = \cos (θ / 2) + i \sin (θ / 2) (σ_{x} \cos α + σ_{y} \cos β + σ_{z} \cos γ) \end{matrix}

Thus the product of the two rotation operators

\begin{matrix} (41.5) & R R^{- 1} = R^{- 1} R = 1 \end{matrix}

is an operator, the unit operator, that leaves unchanged everything that it acts on. The reciprocal

R^{- 1}

of the combination

R = R_{2} R_{1}

of two rotations is

\begin{matrix} (') & R^{- 1} = R_{1}^{- 1} R_{2}^{- 1} \end{matrix}

(reverse order of factors!), as one verifies by substitution into (41.5).

Geometric reason that half angles appear in rotation operators

Algebraic properties of rotation operators

Figure 41.3.
Composition of two rotations seen in terms of reflections. The first rotation (for instance,

90^{\circ}

about

O Z

in the example of Figure 41.1.) is represented in terms of reflection 1 followed by reflection 2 (the planes of the two reflections being separated by

90^{\circ} / 2 = 45^{\circ}

in the example). The second reflection appears as the resultant of reflections 3 and 4. But the reflections 2 and 3 take place in the common plane ZOX. Therefore one reflection undoes the other. Thus the sequence of four operations 1234 collapses to the two reflections 1 and 4 . Their place in turn is taken by a single rotation about the axis

O A

The conjugate transpose,

M^{*}

, of a matrix

M

is obtained by taking the conjugate complex of every element in the matrix and then interchanging rows and columns. By direct inspection of matrix expressions (41.1) one sees that

σ_{x}^{*} = σ_{x}, σ_{y}^{*} = σ_{y}

σ_{z}^{*} = σ_{z}

. Such matrices are said to be Hermitian. The conjugate transpose of the product

M = P Q

of two matrices is the product

M^{*} = Q^{*} P^{*}

of the individual conjugate transposed matrices taken in the reverse order. For the rotation matrix written down above, note that

R^{*} = R^{- 1}

. Such a matrix is said to be unitary. The

Figure 41.4.
Law of composition of rotations epitomized by a spherical triangle in which each of the three important angles represents half an angle of rotation.
determinant of a unitary matrix may be seen to have absolute value unity from the following line of argument:

\begin{aligned} 1 & = \det (unit matrix) = \det (R R^{- 1}) \\ = \det (R R^{*}) = \det R \det R^{*} \\ (41.6) & = | \det R |^{2} . \end{aligned}

In actuality the determinant of the rotation spin matrix is necessarily unity ("unimodular matrix") as shown in the following exercises

Exercise 41.1. ELEMENTARY FEATURES OF THE ROTATION MATRIX

EXERCISES

Write equation (41.3) in the form

R (θ) = \cos (θ / 2) - i \sin (θ / 2) (σ \cdot n),

and establish the following properties:
(a)

(σ \cdot n)^{2} = 1 \equiv unit matrix;

(b)

tr (σ \cdot n) = 0

(tr means "trace," i.e., sum of diagonal elements);
(c)
(d)

\begin{matrix} \underset{[commutator]}{\underset{⏟}{[R, (σ \cdot n)]}} \equiv R (σ \cdot n) - (σ) R = 0; \\ (41.7) & \frac{d R}{d θ} = - \frac{i}{2} (σ \cdot n) R . \end{matrix}

[Note that if one thinks of

θ

as increasing with angular velocity

ω

, so

d θ / d t = ω =

constant, then this last equation reads

\begin{matrix} (41.7') & \frac{d R}{d t} = - \frac{i}{2} (σ \cdot ω) R \end{matrix}

where

ω = ω n

Exercise 41.2. ROTATION MATRIX HAS UNIT DETERMINANT

Recall from exercise 5.5 that for any matrix

M

one has

d [\ln (\det M)] = tr (M^{- 1} d M)

and use this to show that

\det R

in (41.7) is constant, and therefore equal to

(\det R)_{θ = 0} = 1

Infinitesimal rotations

Representation of a 3-vector as a spin matrix

§41.2. INFINITESIMAL ROTATIONS

A given rotation can be obtained by performing in turn two rotations of half the magnitude, or four rotations of a fourth the magnitude, or eight of an eighth the magnitude, and so on. Thus one arrives in the limit at the concept of an infinitesimal rotation described by the spin matrix

R = 1 - (i / 2) (σ_{x} d θ_{y z} + σ_{y} d θ_{z x} + σ_{z} d θ_{x y})

\begin{matrix} (41.8) & R = 1 - (i d θ / 2) (σ \cdot n) \end{matrix}

Here the quantities

\begin{aligned} d θ_{y z} & = - d θ_{z y} = n^{x} d θ = \cos α d θ, \\ (41.9) & d θ_{z x} & = - d θ_{x z} = n^{y} d θ = \cos β d θ, \\ d θ_{x y} & = - d θ_{y x} = n^{z} d θ = \cos γ d θ, \end{aligned}

are the components of the infinitesimal rotation in the three indicated planes. An infinitesimal rotation in the

(x, y)

-plane through the angle

d θ_{x y}

transforms the vector

x = (x, y, z)

into a new vector with changed components

x^{'}

and

y^{'}

but with unchanged component

z^{'} = z

. More generally, the infinitesimal rotation (41.8) considered in this same "active" sense* produces the transformation

x ⟶ x^{'}

with

\begin{matrix} x^{'} = x - (d θ_{x y}) y - (d θ_{x z}) z \\ (41.10) & y^{'} = - (d θ_{y x}) x + y - (d θ_{y z}) z \\ z^{'} = - (d θ_{z x}) x - (d θ_{z y}) y + z \end{matrix}

Spinor calculus provides an alternative (and shorthand!) means to calculate the foregoing effect of a rotation on a vector. Associate with the vector

x

the spin matrix

\begin{matrix} (41.11) & X = x σ_{x} + y σ_{y} + z σ_{z} = (x \cdot σ) \end{matrix}

and with the vector

x^{'}

a corresponding spin matrix or quaternion

X^{'}

. Then the effect of the rotation is summarized in the formula
$$

\begin{matrix} (41.12) & X ⟶ X^{'} = R X R^{*} \end{matrix}

Test this formula for the general infinitesimal rotation (41.10). It reads

(x^{'} \cdot σ) = [1 - (i d θ / 2) (σ \cdot n)] (x \cdot σ) [1 + (i d θ / 2) (σ \cdot n)]

or, to the first order in the quantity

d θ

\begin{matrix} (41.13) & (x^{'} \cdot σ) = (x \cdot σ) + (i d θ / 2) [(x \cdot σ) (σ \cdot n) - (σ \cdot n) (x \cdot σ)] . \end{matrix}

The product of spin matrices

A = (a \cdot σ)

and

B = (b \cdot σ)

built from two distinct vectors

a

and

b

A B = (a \cdot σ) (b \cdot σ) = a^{x} b^{x} σ_{x}^{2} + a^{x} b^{y} σ_{x} σ_{y} + \dots,

or, according to (41.2),

\begin{matrix} (41.14) & A B = (a \cdot b) + i (a \times b) \cdot σ \end{matrix}

Employ this formula to evaluate the right-hand side of (41.13). In the square brackets, the terms in

(x \cdot n)

have opposite signs and cancel. In contrast, the terms in

(n \times x)

have the same sign. They combine to cancel the factor 2 in

(d θ / 2)

. End up with

(x^{'} \cdot σ) = (x \cdot σ) + d θ (n \times x) \cdot σ

\begin{matrix} (41.15) & x^{'} = [1 + (d θ) n \times] x \end{matrix}

in agreement with (41.10), as was to be shown.
A finite rotation about a given axis can be considered as the composition of infinitesimal rotations about that axis. To see this composition in simplest form, rewrite the spin matrix (41.8) associated with the general infinitesimal rotation as

\begin{matrix} (41.16) & R (d θ) = e^{- (i d θ / 2) (σ \cdot n)} \end{matrix}

(exponential function defined by its power-series expansion). Note that (

σ \cdot n

) commutes (a) with unity and (b) with itself, and in addition (c) has a unit square. Therefore the calculation of the exponential function proceeds no differently here, for spin matrices, than for everyday algebra. The composition of the spin matrices for infinitesimal rotations about an unchanging axis proceeds by adding exponents, to give

\begin{matrix} (41.17) & R (θ) = e^{- i (θ / 2) (σ \cdot n)} \end{matrix}

which can also be obtained immediately from equation (41.7). This expression can be put in another form by developing the power series; thus,

Composition of finite rotation from infinitesimal rotations

Rotation of a 3-vector described in spin-matrix language

\begin{aligned} R (θ) & = \sum_{p = 0}^{\infty} (1 / p!) (- i θ σ \cdot n / 2)^{p} \\ (41.18) & = \sum_{even p} (1 / p!) (- i θ / 2)^{p} + (σ \cdot n) \sum_{odd p} (1 / p!) (- i θ / 2)^{p} \\ = \cos (θ / 2) - i \sin (θ / 2) (σ \cdot n) \end{aligned}

in agreement with the expression (41.3) originally given for a spinor transformation. The effect of one infinitesimal rotation after another after another . . . on a vector is given by

X^{'} = R (d θ) \dots R (d θ) X R^{*} (d θ) \dots R^{*} (d θ)

with the consequence that even for a finite rotation

R = R (θ)

one is correct in employing the formula

\begin{matrix} (41.19) & X^{'} = R X R^{*} \end{matrix}

EXERCISE

Exercise 41.3. MORE PROPERTIES OF THE ROTATION MATRIX

Show that for

X = x \cdot σ

one has the commutation relation

[(σ \cdot n), X] = 2 i (n \times x) \cdot σ

Use this to obtain, from equation (41.19) in the form

X = R X_{0} R^{*}

[where

X_{0}

is constant, while

R (θ)

is given by equation (41.17)], the formula

\frac{d}{d θ} (x \cdot σ) = (n \times x) \cdot σ .

Why is this equivalent to the standard definition

\frac{d x}{d t} = ω \times x

for the angular velocity? Reverse the argument to show that equation (41.7 ) correctly defines the rotation

R (t)

resulting from a time-dependent angular velocity

ω (t)

, even though the simple solution

R = \exp [- \frac{1}{2} i t (σ \cdot ω)]

of this equation can no longer be written when

ω

is not constant.

4-vectors and Lorentz transformations in spin-matrix language

§41.3. LORENTZ TRANSFORMATION VIA SPINOR ALGEBRA

Generate a rotation by two reflections in space? Then why not generate a Lorentz transformation by two reflections in spacetime? If for this purpose one has to turn from a real half-angle between the two planes of reflection to a complex half-angle, that development will come as no surprise; nor will it be a surprise that one can
still represent the effect of the Lorentz transformation by a matrix multiplication of the form

\begin{matrix} (41.20) & X ⟶ X^{'} = L X L^{*} \end{matrix}

Here the "Lorentz spin transformation matrix"

L

is a generalization of the rotation matrix,

R

. Also the "coordinate-generating spin matrix"

X

is now generalized from (41.11) to

\begin{matrix} (41.21) & X = t + (x \cdot σ) \end{matrix}

\begin{matrix} (41.22) & X = ‖ \begin{array}{cc} t + z & x - i y \\ x + i y & t - z \end{array} ‖ \end{matrix}

It is demanded that this matrix be Hermitian

\begin{matrix} (41.23) & X = X^{*} \end{matrix}

Then and only then are the coordinates

(t, x, y, z)

real. The conjugate transpose of the transformed spin matrix must also be Hermitian-and is:

\begin{aligned} {(X^{'})}^{*} & = {(L X L^{*})}^{*} \\ (41.24) & = {(L^{*})}^{*} (X)^{*} (L)^{*} = L X L^{*} = X^{'} . \end{aligned}

Therefore the new coordinates

(t^{'}, x^{'}, y^{'}, z^{'})

are guaranteed to be real, as desired. This reality requirement is a rationale for the form of the spin-matrix transformation (41.20), with

L

appearing on one side of

X

and

L^{*}

on the other.

A Lorentz transformation is defined by the circumstance that it leaves the interval invariant:

\begin{matrix} (41.25) & t^{' 2} - x^{' 2} - y^{' 2} - z^{' 2} = t^{2} - x^{2} - y^{2} - z^{2} \end{matrix}

Note that the determinant of the matrix

X

as written out above has the value

\begin{matrix} (41.26) & \det X = t^{2} - x^{2} - y^{2} - z^{2} \end{matrix}

Consequently the requirement for the preservation of the interval may be put in the form

\begin{matrix} (41.27) & \det X^{'} = \det X \end{matrix}

\begin{matrix} (41.28) & (\det L) (\det X) (\det L^{*}) = \det X \end{matrix}

This requirement is fulfilled by demanding

\begin{matrix} (41.29) & \det L = 1 \end{matrix}

[it is not a useful generalization to multiply every element of

L

here by a common phase factor

e^{i δ}

, and therefore multiply det

L

e^{2 i δ}

, because the net effect of this phase factor is nil in the formula

X^{'} = L X L^{*}]

Infinitesimal Lorentz transformations

Composition of finite Lorentz transformations from infinitesimal transformations

The spin matrix associated with a rotation, whether finite or infinitesimal, already satisfied the condition

\det L = 1

[proved in exercise (41.2)]. This condition, being algebraic, will continue to hold when the real angles

d θ_{y z}, d θ_{z x}, d θ_{x y}

, are replaced by complex angles,

d θ_{y z} + i d α_{x}, d θ_{z x} + i d α_{y}, d θ_{x y} + i d α_{z}

. The spin-transformation matrix acquires in this way a total of six parameters, as needed to describe the general infinitesimal Lorentz transformation. Thus the spin matrix for the general infinitesimal Lorentz transformation can be put in the form

\begin{aligned} L = & 1 - (i / 2) (σ_{x} d θ_{y z} + σ_{y} d θ_{z x} + σ_{z} d θ_{x y}) \\ (41.30) & + (1 / 2) (σ_{x} d α_{x} + σ_{y} d α_{y} + σ_{z} d α_{z}) \\ = & 1 - (i d θ / 2) (σ \cdot n) + (σ \cdot d α / 2) \end{aligned}

The effect of this transformation upon the coordinates is to be read out from the formula

X ⟶ X^{'} = L X L^{*}

\begin{aligned} t^{'} + (σ \cdot x^{'}) = & [1 - (i d θ / 2) (σ \cdot n) + (σ \cdot d α / 2)] \\ (41.31) & \times [t + (σ \cdot x)] [1 + (i d θ / 2) (σ \cdot n) + (σ \cdot d α / 2)] \end{aligned}

Employ equation (41.14) for

(σ \cdot A) (σ \cdot B)

to reduce the right side to the form

t + (σ \cdot x) + (σ \cdot d α) t + d θ (n \times x) \cdot σ + (x \cdot d α)

Now compare coefficients of

1, σ_{x}, σ_{y}

and

σ_{z}

, respectively, on both sides of the equation, and find

\begin{matrix} (41.32) & t^{'} = t + (x \cdot d α) \\ x^{'} = x + t d α + d θ (n \times x), \end{matrix}

in agreement with the conventional expression for an infinitesimal Lorentz transformation or "boost" of velocity

d α

, in active form, as was to be shown.

The composition of such infinitesimal Lorentz transformations gives a finite Lorentz transformation. The result, however, can be calculated easily only when all infinitesimal transformations commute. Thus assume that

d θ

and

d α

are in a fixed ratio, so

ω \equiv n \frac{d θ}{d τ} and a \equiv \frac{d α}{d τ}

are constants, with

τ

a parameter. Then integration with respect to

τ

(composition of infinitesimal transformations) gives a finite transformation

L = \exp [- \frac{1}{2} i τ σ

(ω + i a)]

. For

τ = 1

, so

θ n = ω τ, α = a τ

, this reads

\begin{matrix} (41.33) & L = \exp [(α - i θ n) \cdot σ / 2] \end{matrix}

In the special case of a pure boost (no rotation;

θ = 0

), the exponential function is evaluated along the lines indicated in (41.18), with the result

\begin{matrix} (41.34) & L = \cosh (α / 2) + (n_{α} \cdot σ) \sinh (α / 2) \end{matrix}

Here

n_{α} = α / α

is a unit vector in the direction of the boost. The corresponding Lorentz transformation itself is evaluated from the formula

X^{'} = L X L^{*}

\begin{aligned} t^{'} + (x^{'} \cdot σ) = & [\cosh α / 2 + (n_{α} \cdot σ) \sinh α / 2] [t + (x \cdot σ)] \\ (41.35) & \times [\cosh α / 2 + (n_{α} \cdot σ) \sinh α / 2] . \end{aligned}

Simplify with the help of the relations

\begin{matrix} \cosh^{2} (α / 2) + \sinh^{2} (α / 2) = \cosh α \\ 2 \sinh (α / 2) \cosh (α / 2) = \sinh α \end{matrix}

and

(n_{α} \cdot σ) (x \cdot σ) (n_{α} \cdot σ) = 2 (n_{α} \cdot x) (n_{α} \cdot σ) - (n_{α} \cdot n_{α}) (x \cdot σ),

and on both sides of the equation compare coefficients of 1 and

σ

, to find

\begin{aligned} t^{'} & = (\cosh α) t + (\sinh α) (n_{α} \cdot x), \\ (41.36) & x^{'} & = [(\sinh α) n_{α} t + (\cosh α) (n_{α} \cdot x) n_{α}] ("in-line part of transformation") \\ + [x - (x \cdot n_{α}) n_{α}] ("perpendicular part of x unchanged"). \end{aligned}

In this way one verifies that the quantity

α

is the usual "velocity parameter," connected with the velocity itself by the relations

\begin{aligned} {(1 - β^{2})}^{- 1 / 2} & = \cosh α \\ (41.37) & β {(1 - β^{2})}^{- 1 / 2} & = \sinh α \\ β & = \tanh α . \end{aligned}

That velocity parameters add for successive boosts in the same direction shows nowhere more clearly than in the representation (41.33) of the spin-transformation matrix:

\begin{aligned} L (α_{2}) L (α_{1}) & = \exp [α_{2} (n_{α} \cdot σ) / 2] \exp [α_{1} (n_{α} \cdot σ) / 2] = \exp [(α_{2} + α_{1}) (n_{α} \cdot σ) / 2] \\ (41.38) & = L (α_{2} + α_{1}) . \end{aligned}

Turn from this special case, and ask how to get the resultant of two arbitrary Lorentz transformations, each of which is a mixture of a rotation and a boost. No simpler method offers itself to answer this question than to use formula (41.33) together with the equation

\begin{matrix} (41.39) & L (resultant) = L_{2} L_{1} . \end{matrix}

§41.4. THOMAS PRECESSION VIA SPINOR ALGEBRA

A spinning object, free of all torque, but undergoing acceleration, changes its direction as this direction is recorded in an inertial frame of reference. This is the

Thomas precession [see exercise 6.9 and first term in equation (40.33b)]. This precession accounts for a factor two in the effective energy of coupling of spin and orbital angular momentum of an atomic electron. In a nucleus it contributes a little to the coupling of the spin and orbit of a nucleon. The evaluation of the Thomas precession affords an illustration of spin-matrix methods in action.

The precession in question can be discussed quite without reference either to angular momentum or to mass in motion. It is enough to consider a sequence of inertial frames of reference

S (t)

with these two features. (1) To whatever point the motion has taken the mass at time

t

, at that point is located the origin of the frame

S (t)

. (2) The inertial frame

S (t + d t)

at the next succeeding moment has undergone no rotation with respect to the inertial frame

S (t)

, as rotation is conceived by an observer in that inertial frame. However, it has undergone a rotation ("Thomas precession") as rotation is conceived and defined in the laboratory frame of reference.

How is it possible for "no rotation" to appear as "rotation"? The answer is this: one pure boost, followed by another pure boost in another direction, does not have as net result a third pure boost; instead, the net result is a boost plus a rotation. This idea is not new in kind. Figure 41.1 illustrated how a rotation about the

z

-axis followed by a rotation about the

x

-axis had as resultant a rotation about an axis with not only an

x

-component and a

z

-component but also a

y

-component. What is true of rotations is true of boosts: they defy the law for the addition of vectors.

Let the frame

S_{0}

coincide with the laboratory frame, and let the origin of this laboratory frame be where the moving frame is at time

t

. Let

S (t)

be a Lorentz frame moving with this point at time

t

. Let one pure boost raise its velocity relative to the laboratory from

β

β + d β

. The resulting final configuration cannot be reached from

S_{0}

by a pure boost. Instead, first turn

S_{0}

relative to the laboratory frame ("rotation

R

associated with the Thomas precession") and then send it by a simple boost to the final configuration. Only one choice of this rotation will be right to produce match-up. Thus, distinguishing the spin matrices for pure boosts and pure rotations by the letters

B

and

R

, one has the relation

\begin{matrix} (41.40) & B (β + d β) R (ω d t) = " B (d β) " B (β) \end{matrix}

out of which to find the angular velocity

ω

of the Thomas precession. The quotation marks in "

B (d β)

" carry a double warning: (1) the velocity of transformation that boosts

S (t)

S (t + d t)

is not

(β + d β) - β = d β

(law of vector addition-or subtraction-not applicable to velocity), and (2) "

B (d β)

" does not appear as a pure boost in the laboratory frame. It appears as a pure boost only in the comoving frame.
Take care of the second difficulty first. It is only a difficulty because the formalism for combination of transformations,

R_{3} = R_{2} R_{1}

, as developed in

§ 41.1

presupposes all operations

R_{1}, R_{2}, \dots

, to be defined and carried out in the laboratory reference frame. In contrast, the quantity "

B (d β)

" is understood to imply a pure boost as defined and carried out in the comoving frame. Such an operation can be fitted into the formalism as follows. (1) Undo any velocity that the object already has. In other words apply the operator

B^{- 1} (β)

. Then the object is at rest in the laboratory frame. Then apply the necessary small pure boost,

B (a_{comoving} d τ)

, where

a_{comoving}

is the acceleration as it will be sensed by the object and

d τ

is the lapse of proper time as it will be sensed by the object. At the commencement of this brief acceleration the object is at rest relative to the laboratory. What is a pure boost to it is a pure boost relative to the laboratory. It is also a pure boost in the spin-matrix formalism. Then transform back from laboratory to moving frame. Thus have the relation

\begin{matrix} (41.41) & " B (d β) " = B (β) B (a_{comoving} d τ) B^{- 1} (β) \end{matrix}

The equation for the determination of the Thomas precession now reads

\begin{matrix} (41.42) & B (β + d β) R (ω d t) = B (β) B (a_{comoving} d τ) \end{matrix}

or, with all unknowns put on the left,

\begin{matrix} (41.43) & R (ω d t) B^{- 1} (a_{comoving} d τ) = B^{- 1} (β + d β) B (β) . \end{matrix}

The first task, to replace the erroneous value of the velocity change

(d β)

by a correct value (

a_{comoving} d τ

), is now made part of the problem along with the evaluation of the Thomas precession itself.

Principles settled, the calculation proceeds by inserting the appropriate expressions for all four factors in (41.43), and evaluating both sides of the equation to the first order of small quantities, as follows:

\begin{aligned} 1 - (i d t ω + d τ a) \cdot σ / 2 = & [\cosh (α^{'} / 2) - (n_{α^{'}} \cdot σ) \sinh (α^{'} / 2)] \\ (41.44) & \times [\cosh (α / 2) + (n_{α} \cdot σ) \sinh (α / 2)] . \end{aligned}

Here

α

and

n_{α}

are the velocity parameter and unit vector that go with the velocity

β; α^{'} = α + d α

, and

n_{α^{'}} = n_{α} + d n_{α}

, go with

β + d β

. Develop the righthand side of (41.44) by the methods of calculus, writing

α^{'} = α + d α

and

n_{α^{'}} = n_{α} + d n_{α}

, and applying the rule for the differentiation of a product. Equate coefficients of

- σ / 2

and

- i σ / 2

on both sides of the equation. Thus find

\begin{matrix} (41.45) & a_{comoving} d τ = (d α) n_{α} + (\sinh α) d n_{α} \end{matrix}

and

\begin{matrix} (41.46) & ω d t = [2 \sinh^{2} (α / 2)] d n_{α} \times n_{α} . \end{matrix}

The one expression gives the change of velocity as seen in a comoving inertial frame. The other gives the precession as seen in the laboratory frame. For low velocities the expression for the Thomas precession reduces to

\begin{matrix} (41.47) & ω = a \times β / 2 . \end{matrix}

Here

a

is the acceleration. Only the component perpendicular to the velocity

β

is relevant for the precession.

For an elementary account of the importance of the Thomas precession in atomic physics, see, for example, Ruark and Urey (1930).

Angular velocity of Thomas precession
"Orientation-entanglement relation" between a cube and the walls of a room. A

360^{\circ}

rotation of the cube entangles the threads. A

720^{\circ}

rotation might be thought to entangle them still more-but instead makes it possible completely to disen-

§41.5. SPINORS

Orientation-entanglement relation

Figure 41.5. tangle them.

Paint each face of a cube a different color. Then connect each corner of the cube to the corresponding corner of the room with an elastic thread (Figure 41.5). Now rotate the cube through

2 π = 360^{\circ}

. The threads become tangled. Nothing one can do will untangle them. It is impossible for every thread to proceed on its way in a straight line. Now rotate the cube about the same axis by a further

2 π

. The threads become still more tangled. However, a little work now completely straightens out the tangle (Figure 41.6). Every thread runs as it did in the beginning in a straight line from its corner of the cube to the corresponding corner of the room. More generally, rotations by

0, \pm 4 π, \pm 8 π, \dots

, leave the cube in its standard "orienta-tion-entanglement relation" with its surroundings, whereas rotations by

\pm 2 π, \pm 6 π

\pm 10 π, \dots

, restore to the cube only its orientation, not its orientation-entanglement relation with its surroundings. Evidently there is something about the geometry of orientation that is not fully taken into account in the usual concept of orientation; hence the concept of "orientation-entanglement relation" or (briefer term!) "version" (Latin versor, turn). Whether there is also a detectable difference in the physics (contact potential between a metallic object and its metallic surroundings, for example) for two inequivalent versions of an object is not known [Aharonov and Susskind (1967)].

In keeping with the distinction between the two inequivalent versions of an object, the spin matrix associated with a rotation,

\begin{matrix} (41.48) & R = \cos (θ / 2) - i (n \cdot σ) \sin (θ / 2), \end{matrix}

reverses sign on a rotation through an odd multiple of

2 π

. This sign change never shows up in the law of transformation of a vector, as summarized in the formula

\begin{matrix} (41.49) & X ⟶ X^{'} = R X R^{*} \end{matrix}

(two factors

R

; sign change in each!). The sign change does show up when one turns from a vector to a 2 -component quantity that transforms according to the law

\begin{matrix} (41.50) & ξ ⟶ ξ^{'} = R ξ . \end{matrix}

Figure 41.6.
An object is connected to its surroundings by elastic threads as in Figure 41.5. (Eight are shown here; any number could be used.) Rotating the object through

720^{\circ}

and then following the procedure outlined (Edward McDonald) in frames 2-8 (with the object remaining fixed), one finds that the connecting threads are left disentangled, as in frame 9 (lower right).

Such a quantity is known as a spinor. A spinor reverses sign on a

360^{\circ}

rotation. It therefore provides a reasonable means to keep track of the difference between the two inequivalent versions of the cube. More generally, with each orientationentanglement relation between the cube and its surroundings one can associate a different value of the spinor

ξ

. Moreover, there is nothing that limits the usefulness of the spinor concept to rotations. Also, for the general combination of boost and rotation, one can write

\begin{matrix} (41.51) & ξ ⟶ ξ^{'} = L ξ \end{matrix}

Lorentz transformation of a spinor

When the boost and rotation are both infinitesimal, the explicit form of this transformation is simple:

ξ^{'} = [1 - (i d θ / 2) (n \cdot σ) + (d β / 2) \cdot σ] ξ,

or, according to (41.1),

\begin{matrix} (41.52) & (\binom{ξ^{'}}{ξ^{' 2}}) = ‖ \begin{array}{ll} 1 + \frac{1}{2} (- i θ_{x y} + β_{z}) & \frac{1}{2} (- i θ_{y z} - θ_{z x} + β_{x} - i β_{y}) ‖ (\begin{matrix} ξ^{1} \\ \frac{1}{2} (- i θ_{y z} + θ_{z x} + β_{x} + i β_{y}) \\ ξ^{2} \end{matrix}), \frac{1}{2} (i θ_{x y} - β_{z}) \end{array} ‖ \end{matrix}

For any combination of a boost in the

z

-direction of any magnitude and a finite rotation about the

z

-axis, one has

\begin{matrix} (41.53) & (\binom{ξ^{''}}{ξ^{' 2}}) = ‖ \begin{array}{cc} e^{- 1 / 2 i θ_{x ν} + 1 / 2 β_{z}} & 0 \\ 0 & e^{1 / 2 i θ_{x y} - 1 / 2 β_{z}} \end{array} ‖ (\binom{ξ^{1}}{ξ^{2}}) . \end{matrix}

To keep track of the two components of the spinor, it is convenient and customary to introduce a label (capital Roman letter near beginning of alphabet) that takes on the values 1 and 2 ; thus (41.51) becomes

\begin{matrix} (41.54) & ξ^{' A} = L^{A}_{B} ξ^{B} . \end{matrix}

The spinor has acquired a significance of its own through one's having pulled out half of the transformation formula

\begin{matrix} (41.55) & X^{'} = L X L^{*} \end{matrix}

To be able to recover this formula, one requires the other half as well. It contains the conjugate complex of the Lorentz transformation. Therefore introduce another

Second type of spinor

Vector regarded as a Hermitian second-rank spinor spinor

η

that transforms according to the law

\begin{matrix} (41.56) & η^{' \dot{U}} = {\bar{L}}_{\dot{U}}^{\dot{U}} η^{\dot{V}} \end{matrix}

[\dot{U} = \dot{1}, \dot{2}; \dot{V} = \dot{1}, 2;

dots and capital letters near the end of the alphabet are used to distinguish components that transform according to the conjugate complex (no transpose!) of the Lorentz spin matrix].

§41.6. CORRESPONDENCE BETWEEN VECTORS AND SPINORS

To go back from spinors to vectors, note that the spin matrix

X

in (41.55) has the form

\begin{matrix} (41.57) & X = t + (x \cdot σ) = ‖ \begin{array}{cc} (t + z) & (x - i y) \\ (x + i y) & (t - z) \end{array} ‖ = ‖ \begin{array}{ll} X^{1 i} & X^{1 \dot{1}} \\ X^{2 i} & X^{2 \dot{2}} \end{array} ‖, \end{matrix}

where the labels receive dots or no dots according as they are coupled in (41.55) to

L^{*}

or to

L

. That equation of transformation becomes

\begin{matrix} (41.58) & X^{' A \dot{U}} = L^{A}_{B} {\bar{L}}^{\dot{U}} \dot{V} X^{B \dot{V}} \end{matrix}

(transpose obtained automatically by ordering of indices; thus

{\bar{L}}^{\dot{U}} \dot{\dot{V}}

, not

L^{* \dot{U}} \dot{\dot{V}}

). The coefficients in this transformation are identical with the coefficients in the law for the transformation of a "second-rank spinor with one index undotted and the other dotted:"

\begin{matrix} (41.59) & ξ^{' A} η^{' \dot{U}} = L^{A}_{B} {\bar{L}}^{\dot{U}} \dot{\dot{V}} {\dot{ξ}}^{β} η^{\dot{η}} \end{matrix}

In this sense one can say that "a 4 -vector transforms like a second-rank spinor." To be completely explicit about this connection between a 4 -vector and a second-rank spinor, note from (41.57) the relations

\begin{aligned} X^{1 \dot{1}} = x^{0} + x^{3}, \\ X^{12} = x^{1} - i x^{2}, \\ (41.60) & X^{2 \dot{1}} = x^{1} + i x^{2}, \\ X^{2 \dot{2}} = x^{0} - x^{3} . \end{aligned}

In a more compact form, one has

\begin{matrix} (41.61) & X^{A \dot{U}} = [t + (x \cdot σ)]^{A \dot{U}} = x^{μ} σ_{μ}^{A \dot{U}} \end{matrix}

where

σ_{0}

is the unit matrix. This equation tells immediately how to go from the components of a 4 -vector, or " 1 -index tensor," to the components of the corresponding " 1,1 -spinor" (one undotted and one dotted index).

With each real 4 -vector

x^{α}

is associated a 1,1 -spinor that is Hermitian in the sense that

\begin{matrix} (41.62) & X^{A \dot{U}} = \overset{―}{X^{U A}} \end{matrix}

An example of a Hermitian 1,1-spinor is provided by (41.61). The concept of Hermiticity can be stated in other words, and more generally. Associated with any

N, N

-spinor

Φ

with components

Φ_{1} A_{1} \dots A_{N} {\dot{U}}_{1} \dots {\dot{θ}}_{N v}

is the conjugate complex spinor

\bar{ϕ}

with

\begin{matrix} (41.63) & (\bar{Φ})^{A_{1} \dots A_{N} {\dot{U}}_{1} \dots {\dot{U}}_{N}} = \overset{―}{(Φ^{U_{1} \dots U_{s} {\dot{i}}_{1} \dots {\dot{i}}_{x}})} \end{matrix}

N, N

-spinor is said to be Hermitian when it is equal to its conjugate complex.

§41.7. SPINOR ALGEBRA

Equation (41.53) showed the component

ξ^{' 1}

of a spinor rising exponentially with a boost in proportion to the factor

e^{1 / 2 β_{z}}

, and the other component,

ξ^{' 2}

falling exponentially. If from two spinors

ξ

and

ζ

, there is to be any quantity constructed

Spinor algebra:

N, N

-spinors and Hermiticity
(1)

ε^{A B}, ε_{A B}

defined
(2) raising and lowering spinor indices
(3) scalar products of spinors
(4) the mapping between vectors and 1,1-spinors
(5)

σ^{μ}

defined and related to

σ_{μ}

ξ^{1} ζ^{2}

and

ξ^{2} ζ^{1}

. One can restate this product prescription in other language. Introduce the alternating symbols

ε^{A B}

and

ε_{A B}

such that

ε^{12} = ε_{12} = 1

and

\begin{matrix} (41.64) & ε^{A B} = - ε^{B A}, ε_{A B} = - ε_{B A} \end{matrix}

the only other nonvanishing components being

ε^{21} = ε_{21} = - 1

. Define the lowerlabel spinor

ξ_{A}

in terms of the upper-label spinor

ξ^{A}

by the equation

\begin{matrix} (41.65) & ξ_{A} = ξ^{B} ε_{B A} \end{matrix}

with the inverse

\begin{matrix} (41.66) & ξ^{B} = ε^{B C} ξ_{C} \end{matrix}

Then the scalar product of one spinor by another is defined to be

\begin{matrix} (41.67) & ξ_{A} ζ^{A} \end{matrix}

The value of this scalar product is unaffected by any boost or rotation or combination thereof:

\begin{aligned} ξ_{A}^{'} ζ^{' A} & = ξ^{' B} ε_{B A} ζ^{' A} \\ = (L^{B}_{D} ξ^{D}) ε_{B A} (L^{A}_{c} ζ^{C}) \\ = (\det L) ξ^{D} ε_{D C} ζ^{C} \\ (41.68) & = ξ_{C} ζ^{C} \end{aligned}

The proof uses the fact that the expression

L^{B}_{D} ε_{B A} L_{C}^{A} (1)

vanishes when

D = C

, and (2) reduces to the determinant of

L

(unity!) or its negative when

D = 1, C = 2

, or

D = 2, C = 1

. Note that the scalar product

ξ^{A} ζ_{A}

is the negative of the scalar product

ξ_{A} ζ^{A}

. The value of the scalar product of a spinor with itself is automatically zero ("built-in null character of a spinor").

The components of a vector with upper index have been expressed in terms of the components of a 1,1 -spinor with upper indices

\begin{matrix} (41.69) & X^{A \dot{U}} = x^{μ} σ_{μ}^{A \dot{U}} \end{matrix}

and a similar correlation holds between vector and 1,1-spinor with lower indices; thus,

\begin{matrix} (41.70) & X_{A \dot{U}} = x_{μ} σ_{A \dot{U}} \end{matrix}

Here the "associated basic spin matrices" have the components

\begin{matrix} (41.71) & σ_{A \dot{U}}^{μ} = η^{μ ν} σ_{ν}^{B \dot{V}} ε_{B A} ε_{\dot{V} \dot{U}} \end{matrix}

or, explicitly,

\begin{matrix} (41.72) & ‖ \begin{array}{ll} σ^{μ}_{1 i} & σ^{μ}_{12} \\ σ^{μ}_{2 i} & σ^{μ}_{2 \dot{2}} \end{array} ‖ = {\begin{cases} - ‖ \begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array} ‖ for μ = 0 \\ - ‖ \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} ‖ for μ = 1 \\ + ‖ \begin{array}{cc} 0 & - i \\ i & 0 \end{array} ‖ for μ = 2 \\ - ‖ \begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array} ‖ for μ = 3 \end{cases} \end{matrix}

The same type of multiplication law holds for these matrices,

{(σ^{x})}^{2} = {(σ^{y})}^{2} =

{(σ^{z})}^{2} = 1, σ^{x} σ^{y} = - σ^{y} σ^{x} = i σ^{z}

, etc., as for the matrices

σ_{x}, σ_{y}, σ_{z}

of (41.2). Between the "basic spin matrices,"

σ_{μ}

, and the "associated basic spin matrices,"

σ^{μ}

, the following orthogonality and normalization relations obtain:

\begin{matrix} (41.73) & σ_{μ}^{A {\dot{U}}_{σ}^{μ}}_{B \dot{V}} = - 2 δ_{B}^{A} δ_{V}^{\dot{U}} \end{matrix}

and

\begin{matrix} (41.74) & σ_{μ}^{A {\dot{U}}_{σ}}_{A \dot{U}} = - 2 δ_{μ}^{ν} . \end{matrix}

One can use these relations to "go back from a quantity expressed as a 1,1 -spinor ('spinor equivalent of a vector') to the same quantity expressed directly as a vector (first-rank tensor)." Thus, multiply through (41.61) on both sides by

- \frac{1}{2} σ^{ν}_{A \dot{U}}

, sum over the spinor indices, and employ (41.74) to find the contravariant components of the vector,

\begin{matrix} (41.75) & x^{ν} = - \frac{1}{2} σ_{A \dot{U}}_{A} X^{A \dot{U}} . \end{matrix}

Similarly from (41.70) and (41.73) one finds the covariant components,

\begin{matrix} (41.76) & x_{v} = - \frac{1}{2} σ_{ν}^{A \dot{U}} X_{A \dot{U}} . \end{matrix}

N

-index tensor

T

lets itself be expressed in spinor language ("spinor equivalent of the tensor") by a generalization of (41.61) or (41.70); thus, for a mixed tensor of third order, one has

\begin{matrix} (41.77) & T_{A \dot{U}}^{B \dot{V} \dot{W}} = σ^{α}_{A \dot{U}}_{β}^{B \dot{V}} {\dot{σ}}_{γ}^{c \dot{W}} T_{α}^{β γ} \end{matrix}

and the converse relation

\begin{matrix} (41.78) & T_{α}^{β γ} = {(- \frac{1}{2})}^{3} σ_{α}_{α} {\dot{U}}_{σ^{β}}_{B \dot{V}} σ^{γ}_{C \dot{W}} T_{A \dot{U}}^{B \dot{V} C \dot{W}} . \end{matrix}

Box 41.1 gives the spinor representation of several simple tensors.

Box 41.1 SPINOR REPRESENTATION OF CERTAIN SIMPLE TENSORS IN THE CONTEXT OF A LOCAL LORENTZ FRAME

Quantity

Tensor language

Spinor language

General 4-vector

x^{α}

(four complex numbers)

X^{A U U}

(4 complex numbers)

Real 4-vector (example: 4-momentum)

x^{α} = {\bar{x}}^{α}

(four real numbers)

X^{A \dot{U}} = \overset{―}{(X^{U A})}

(2 real components,

1 distinct complex component)

Null 4-vector

η_{α β} x^{α} x^{β} = 0

\det X^{A \dot{U}} = 0 [

see (41.57)]; hence there exist two spinors

ξ^{A}

and

η^{U}

such that

X^{A U} = ξ^{A} η^{\dot{U}}

Future-pointing real null 4 -vector (such as 4-momentum of a photon)

\begin{aligned} x^{α} = {\bar{x}}^{α} \\ η_{α β} x^{α} x^{β} = 0 \\ x^{0} > 0 \end{aligned}

There exists a spinor

ξ^{A}

(two complex numbers, unique up to a common multiplicative phase factor

e^{i δ})

such that

X^{A U} = ξ^{A} (\bar{ξ})^{\dot{U}}

Past-pointing real null 4 -vector

x^{0} < 0

X^{A} \dot{U}^{d} = - ξ^{A} (\bar{ξ})^{\dot{U}}

Real bivector or 2 -form (such as Maxwell field)

\begin{aligned} F_{[α β]} (subscript implying F_{α β} = \\ - F_{β α}; six distinct real \\ components) \end{aligned}

There exists a symmetric spinor

ϕ_{A B}

(three distinct complex components

ϕ_{11}, ϕ_{12}, ϕ_{22}

) such that

F_{A \dot{U} \dot{B} \dot{V}} = ϕ_{A B^{ε} \dot{U} \dot{V}} + ε_{A B} (\bar{ϕ})_{\dot{U} \dot{V}}

Real 2 -form dual to foregoing real 2-form

^{*} F_{α β} = \frac{1}{2} ε_{α β γ δ} F^{γ δ}

^{*} F_{A U B V} = - i ϕ_{A B}^{ℓ} \dot{V} V + i i_{A B} (\bar{ϕ})_{\dot{U} \dot{V}}

(duality for 2 -form corresponds to multiplication of spinor

ϕ_{A B}

- i

)

Real fourth-order tensor with symmetries of Weyl conformal curvature tensor; that is, with symmetries of Riemann curvature tensor and with additional requirement of vanishing Ricci tensor ("empty space;" "vacuum Riemann tensor")

C_{α β γ δ} = C_{([α β ‖ [γ δ])}

(antisymmetric in first two indices; antisymmetric in last two indices; symmetric against interchange of first pair with second pair)

C^{α}_{[β γ 8]} = 0

(20 algebraically distinct components, as for the Riemann tensor, reduced to 10 by the further vacuum condition:)

C^{α}_{β a δ} = 0

There exists a completely symmetric spinor

ψ_{A B C D}

with five distinct complex components,

ψ_{1111}

ψ_{1112}

ψ_{1122}

ψ_{1222}

ψ_{2222}

such that

C_{A \dot{A} B \dot{B} C \dot{W} D X} =

ψ_{A B C D} ε_{\dot{U} \dot{\tilde{V}} \dot{W} \dot{X}} + ε_{A B^{ε}} ε_{C D} {\bar{ψ}}_{\dot{U} \dot{V} \dot{W} \dot{X}}

Quantity Tensor language Spinor language General 4-vector x^(alpha) (four complex numbers) X^(AUU) (4 complex numbers) Real 4-vector (example: 4-momentum) x^(alpha)= bar(x)^(alpha) (four real numbers) "X^(AU^(˙))= bar((X^(UA))) (2 real components, 1 distinct complex component)" Null 4-vector eta_(alpha beta)x^(alpha)x^(beta)=0 detX^(AU^(˙))=0[ see (41.57)]; hence there exist two spinors xi^(A) and eta^(U) such that X^(AU)=xi^(A)eta^(U^(˙)). Future-pointing real null 4 -vector (such as 4-momentum of a photon) "x^(alpha)= bar(x)^(alpha) eta_(alpha beta)x^(alpha)x^(beta)=0 x^(0) > 0" There exists a spinor xi^(A) (two complex numbers, unique up to a common multiplicative phase factor {:e^(i delta)) such that X^(AU)=xi^(A)( bar(xi))^(U^(˙)) Past-pointing real null 4 -vector x^(0) < 0 X^(A)U^(˙)^("d ")=-xi^(A)( bar(xi))^(U^(˙)). Real bivector or 2 -form (such as Maxwell field) "F_([alpha beta]) (subscript implying F_(alpha beta)= -F_(beta alpha); six distinct real components) " There exists a symmetric spinor phi_(AB) (three distinct complex components phi_(11),phi_(12),phi_(22) ) such that F_(AU^(˙)B^(˙)V^(˙))=phi_(AB^(epsi)U^(˙)V^(˙))+epsi_(AB)( bar(phi))_(U^(˙)V^(˙)) Real 2 -form dual to foregoing real 2-form ^(**)F_(alpha beta)=(1)/(2)epsi_(alpha beta gamma delta)F^(gamma delta) "^(**)F_(AUBV)=-iphi_(AB)^(ℓ)V^(˙)V+ii_(AB)( bar(phi))_(U^(˙)V^(˙)) (duality for 2 -form corresponds to multiplication of spinor phi_(AB) by -i )" Real fourth-order tensor with symmetries of Weyl conformal curvature tensor; that is, with symmetries of Riemann curvature tensor and with additional requirement of vanishing Ricci tensor ("empty space;" "vacuum Riemann tensor") C_(alpha beta gamma delta)=C_(([alpha beta||[gamma delta])) (antisymmetric in first two indices; antisymmetric in last two indices; symmetric against interchange of first pair with second pair) C^(alpha)_([beta gamma8])=0 (20 algebraically distinct components, as for the Riemann tensor, reduced to 10 by the further vacuum condition:) C^(alpha)_(beta a delta)=0 "There exists a completely symmetric spinor psi_(ABCD) with five distinct complex components, psi_(1111) psi_(1112) psi_(1122) psi_(1222) psi_(2222) such that C_(AA^(˙)BB^(˙)CW^(˙)DX)= psi_(ABCD)epsi_(U^(˙) tilde(V)^(˙)W^(˙)X^(˙))+epsi_(AB^(epsi))epsi_(CD) bar(psi)_(U^(˙)V^(˙)W^(˙)X^(˙))"

| Quantity | Tensor language | Spinor language | | :---: | :---: | :---: | | General 4-vector | $x^{\alpha}$ (four complex numbers) | $X^{A U U}$ (4 complex numbers) | | Real 4-vector (example: 4-momentum) | $x^{\alpha}=\bar{x}^{\alpha}$ (four real numbers) | $X^{A \dot{U}}=\overline{\left(X^{U A}\right)}$ (2 real components, 1 distinct complex component) | | Null 4-vector | $\eta_{\alpha \beta} x^{\alpha} x^{\beta}=0$ | $\operatorname{det} X^{A \dot{U}}=0[$ see (41.57)]; hence there exist two spinors $\xi^{A}$ and $\eta^{U}$ such that $X^{A U}=\xi^{A} \eta^{\dot{U}}$. | | Future-pointing real null 4 -vector (such as 4-momentum of a photon) | $\begin{aligned} & x^{\alpha}=\bar{x}^{\alpha} \\ & \eta_{\alpha \beta} x^{\alpha} x^{\beta}=0 \\ & x^{0}>0 \end{aligned}$ | There exists a spinor $\xi^{A}$ (two complex numbers, unique up to a common multiplicative phase factor $\left.e^{i \delta}\right)$ such that $X^{A \mathscr{U}}=\xi^{A}(\bar{\xi})^{\dot{U}}$ | | Past-pointing real null 4 -vector | $x^{0}<0$ | $X^{A} \dot{U}{ }^{\text {d }}=-\xi^{A}(\bar{\xi})^{\dot{U}}$. | | Real bivector or 2 -form (such as Maxwell field) | $\begin{aligned} & F_{[\alpha \beta]} \text { (subscript implying } F_{\alpha \beta}= \\ & -F_{\beta \alpha} \text {; six distinct real } \\ & \text { components) } \end{aligned}$ | There exists a symmetric spinor $\phi_{A B}$ (three distinct complex components $\phi_{11}, \phi_{12}, \phi_{22}$ ) such that $F_{A \dot{U} \dot{B} \dot{V}}=\phi_{A B^{\varepsilon} \dot{U} \dot{V}}+\varepsilon_{A B}(\bar{\phi})_{\dot{U} \dot{V}}$ | | Real 2 -form dual to foregoing real 2-form | ${ }^{*} F_{\alpha \beta}=\frac{1}{2} \varepsilon_{\alpha \beta \gamma \delta} F^{\gamma \delta}$ | ${ }^{*} F_{A U B V}=-i \phi_{A B}{ }^{\ell} \dot{V} V+i i_{A B}(\bar{\phi})_{\dot{U} \dot{V}}$ (duality for 2 -form corresponds to multiplication of spinor $\phi_{A B}$ by $-i$ ) | | Real fourth-order tensor with symmetries of Weyl conformal curvature tensor; that is, with symmetries of Riemann curvature tensor and with additional requirement of vanishing Ricci tensor ("empty space;" "vacuum Riemann tensor") | $C_{\alpha \beta \gamma \delta}=C_{([\alpha \beta \\|[\gamma \delta])}$ (antisymmetric in first two indices; antisymmetric in last two indices; symmetric against interchange of first pair with second pair) $C^{\alpha}{ }_{[\beta \gamma 8]}=0$ (20 algebraically distinct components, as for the Riemann tensor, reduced to 10 by the further vacuum condition:) $C^{\alpha}{ }_{\beta a \delta}=0$ | There exists a completely symmetric spinor $\psi_{A B C D}$ with five distinct complex components, $\psi_{1111}$ $\psi_{1112}$ $\psi_{1122}$ $\psi_{1222}$ $\psi_{2222}$ such that $C_{A \dot{A} B \dot{B} C \dot{W} D X}=$ $\psi_{A B C D} \varepsilon_{\dot{U} \dot{\tilde{V}} \dot{W} \dot{X}}+\varepsilon_{A B^{\varepsilon}} \varepsilon_{C D} \bar{\psi}_{\dot{U} \dot{V} \dot{W} \dot{X}}$ |

In some treatises on spinor analysis, the factor

{(- \frac{1}{2})}^{N}

in equations like (41.78) is eliminated by the following double prescription: (1) insert into the matrices

σ_{μ}

and

σ^{μ}

a factor

1 / \sqrt{2}

not included above; and (2) use for the standard metric not diag

η_{μ ν} = (- 1, 1, 1, 1)

as above, but

(1, - 1, - 1, - 1)

. This prescription was not adopted here (1) because the introduction of

1 / \sqrt{2}

in the matrices

σ_{x}, σ_{y}, σ_{z}

would put them out of line with the Pauli matrices as used for many years throughout

Quantity

Tensor language

Spinor language

§41.8. SPIN SPACE AND ITS BASIS SPINORS

The "space" of elementary spinors is two-dimensional. Therefore it is spanned by any two linearly independent spinors

λ_{A}

and

μ_{A}

. Moreover, it is easy to diagnose a pair of spinors for possible linear dependence, that is, for existence of a relation of the form

μ_{A} =

const

λ_{A}

. In this event, the scalar product of

μ_{A}

with

λ^{A}

, like the scalar product of

λ_{A}

with

λ^{A}

(41.67) automatically vanishes. Therefore a nonvanishing scalar product

\begin{matrix} (41.79) & λ_{A} μ^{A} \neq 0 \end{matrix}

is a necessary and sufficient condition for the linear independence of two spinors.
The general 4-vector lets itself be represented as a linear combination of four basis vectors. Similarly the general spinor lets itself be represented as a linear combination of two basis spinors:

\begin{matrix} (41.80) & ω^{A} = λ ξ^{A} + μ η^{A} . \end{matrix}

Here it is understood that the term "basis spinor" implies that

ξ^{A}

and

η^{A}

satisfy the normalization condition

\begin{matrix} (41.81) & ξ_{A} η^{A} = 1 (= - η_{A} ξ^{4}) \end{matrix}

From this condition one derives simple expressions for the expansion coefficients in (41.80):

\begin{aligned} λ = - η_{A} ω^{A} (= ω_{B} η^{B}), \\ (41.82) & μ = ξ_{A} ω^{A} (= - ω_{B} ξ^{B}) . \end{aligned}

Inserting these expansion coefficients back into (41.80) will reproduce any arbitrarily chosen spinor

ω^{A}

. In other words, the following equation has to be an identity in the components of

ω_{B}

\begin{matrix} (41.83) & ω^{A} = ε^{A B} ω_{B} \equiv (ξ^{A} η^{B} - η^{A} ξ^{B}) ω_{B} \end{matrix}

From this circumstance, it follows that the components of the two basic spinors are linked by the equations

\begin{matrix} (41.84) & ξ^{A} η^{B} - η^{A} ξ^{B} = ε^{A B} . \end{matrix}

Given two basis spinors

ξ^{A}

and

η^{A}

, one can get two equally good basis spinors by writing

\begin{aligned} ξ_{new}^{A} = ξ^{A}, \\ (41.85) & η^{A}_{new} = η^{A} + α ξ^{A}, \end{aligned}

with

α

any real or complex constant, as one checks at once by substitution into (41.81) or (41.84). The most general "spinor mate" to a given spinor

ξ^{A}

, satisfying the normalization condition (41.81), has this form (41.85).

Figure 41.7.
Spinor represented by (1) "flagpole" [Penrose terminology; track of pulse of light; null vector

O P

] plus (2) "flag" [arrow

(P ⟶

) flashed onto surface of moon by laser pulse from earth or, in expanded view in the inset above, a flag itself, substituted for the arrow] plus (3) the orientation-entanglement relation between the flag and its surroundings [symbolized by strings drawn from corners of flag to surroundings]. When the spinor itself is multiplied by a factor

ρ e^{i σ}

, the components of the null vector (flagpole) are multiplied by the factor

ρ^{2}

and the flag is rotated through the angle

2 σ

about the flagpole.

§41.9. SPINOR VIEWED AS FLAGPOLE PLUS FLAG PLUS ORIENTATION-ENTANGLEMENT RELATION

How can one visualize a spinor? Aim the laser, pull the trigger, and send a megajoule pulse from the here and now (event

O

) to the there and then (event

P

: center of the crater Aristarchus,

400, 000 km

from

θ

in space, and

400, 000 km

from

θ

in light-travel time). The laser has been designed to produce, not a mere spot of light, but an illuminated arrow. Following Roger Penrose, speak of the null vector

O P

as a "flagpole," and of the illuminated arrow as a "flag." A spinor (Figure 41.7) consists of this combination of (1) null flagpole plus (2) flag plus (3) the orientation-

Geometric representation of a spinor:
(1) null vector (flagpole), plus
(2) bivector (flag) and its orientation-entanglement relation
entanglement relation between the flag and its surroundings. "Rotate the flag" by repeatedly firing the laser, with a bit of rotation of the laser about its axis between one firing and the next. When the flag has turned through

360^{\circ}

and has come back to its original direction, the spinor has reversed sign. A rotation of the flag about the flagpole through any even multiple of

2 π

restores the spinor to its original value.

One goes from a spinor

ξ

, a mathematical object with two complex components

ξ^{1}

and

ξ^{2}

, to the geometric object of "flagpole plus flag plus orientation-entanglement relation" in two steps: first the pole, then the flag. Thus, go from the spinor

ξ^{A}

to the real null 4 -vector of the "pole" by way of the formula

\begin{matrix} (41.86) & x^{α} ⟶ X^{A \dot{U}} = ξ^{A} (\bar{ξ})^{\dot{U}} \end{matrix}

\begin{matrix} (41.87) & ‖ \begin{array}{ll} (t + z) & (x - i y) \\ (x + i y) & (t - z) \end{array} ‖ = ‖ \begin{array}{cc} ξ^{1} {\bar{ξ}}^{i} & ξ^{1} {\bar{ξ}}^{2} \\ ξ^{2} {\bar{ξ}}^{i} & ξ^{2} {\bar{ξ}}^{2} \end{array} ‖ \end{matrix}

The matrix on the right has its first row identical up to a factor

ξ^{1} / ξ^{2}

with its second row. Therefore the determinant of the matrix on the right vanishes. So also for the left. Therefore the 4 -vector

O P = (t, x, y, z)

is a null vector. One "stretches" this vector by a factor

ρ^{2}

when one multiplies the spinor

ξ^{A}

by the nonzero complex number

λ = ρ e^{i σ}

(

ρ, σ

real); however, the vector is unchanged in direction. The 4 -vector is also unaffected by the choice of the angle

σ

. In other words, this null 4 -vector is uniquely fixed by the spinor; but the spinor is not fixed with all uniqueness by the 4 -vector. To a given 4 -vector corresponds a whole family of spinors. They differ from one another by a multiplicative phase factor of the form

e^{i σ}

("flag factor").

Looking further to see the influence of the flag factor showing up, turn from a real vector (four components) generated out of the spinor

ξ^{A}

to a real bivector (six components) generated out of the same spinor:

\begin{aligned} (41.88) & F^{μ ν} ⟶ & F^{A B \dot{U} \dot{V}} = ξ^{A} ξ^{B} \dot{ϵ} \dot{U} \dot{V} \end{aligned} + ϵ^{A B} (\bar{ξ})^{\dot{U}} (\bar{ξ})^{\dot{V}}, 子; \dot{U}; ν ⟶ B .

That this quantity has no more than six distinct components (

F^{μ ν} = - F^{ν μ}

) follows from interchanging

A

with

B

and

\dot{U}

with

\dot{V}

, and noting the resultant change in sign on the righthand side of (41.88). To unfold the meaning of this bivector, look in (41.88) for every appearance of the alternating factor

ε^{A B}

. Wherever such a factor appears, insert the expression (41.84) for this factor in terms of the starting spinor

ξ^{A}

and insert the additional spinor

η^{A}

that is needed, along with

ξ^{A}

, to supply a basis for all spinors. In this way, find

\begin{aligned} F^{μ ν} ⟶ F^{A B \dot{U} \dot{V}} & = ξ^{A} ξ^{B} ({\bar{ξ}}^{\dot{U}} {\bar{η}}^{\dot{V}} - {\bar{η}}^{\dot{U}} {\bar{ξ}}^{\dot{V}}) + (ξ^{A} η^{B} - η^{A} ξ^{B}) {\bar{ξ}}^{\dot{U}} {\bar{ξ}}^{\dot{V}} \\ (41.89) & = ξ^{A} {\bar{ξ}}^{\dot{U}} (ξ^{B} {\bar{η}}^{\dot{V}} + η^{B} {\bar{ξ}}^{\dot{V}}) - (ξ^{A} {\bar{η}}^{\dot{U}} + η^{A} {\bar{ξ}}^{\dot{U}}) ξ^{B} {\bar{ξ}}^{\dot{V}} \\ = X^{A \dot{U}} Y^{B \dot{V}} - Y^{A \dot{U}} X^{B \dot{V}} ⟶ x^{μ} y^{ν} - y^{μ} x^{ν} \end{aligned}

Thus the 2,2 -spinor built from

ξ^{A}

represents a bivector constructed out of the two 4 -vectors

x

and

y

. Of these, the first is the "real null vector of the flagpole," already seen to be determined uniquely by the spinor

ξ^{A}

. The second vector,

\begin{matrix} (41.90) & y^{α} ⟶ Y^{A \dot{U}} = ξ^{A} \bar{η} \dot{U} + η^{A} \bar{ξ} \dot{U}, \end{matrix}

is also determined by

ξ^{A}

, but not uniquely, because the "spinor mate,"

η^{A}

, to

ξ^{A}

is not unique. Go from one choice of mate,

η^{A}

, to a new choice of mate (equation 41.85),

\begin{matrix} (41.91) & η_{new}^{A} = η^{A} + α ξ^{A} . \end{matrix}

Then the real 4 -vector

y^{μ}

goes to the new real 4 -vector

\begin{matrix} (41.92) & y_{new}^{μ} = y^{μ} + (α + \bar{α}) x^{μ} \end{matrix}

Were the 4 -vector

y

unique, there would project out from the flagpole, not a flag but an arrow. The range of values open for the real constant

α + \bar{α}

makes one arrow into many arrows, all coplanar; hence the flag of Penrose. Otherwise stated, the choice of a spinor

ξ^{A}

fixes no individual arrow, but does fix the totality of the collection of arrows, and thus uniquely specifies the flag.
The 4-vector

y

(and with it

y_{new}

) is orthogonal to the null 4-vector

x

\begin{matrix} (41.93) & \begin{aligned} x \cdot y & = x_{β} y^{β} = - \frac{1}{2} X_{A \dot{U}} Y^{A \dot{U}} \\ = - \frac{1}{2} ξ_{A} {\bar{ξ}}_{\dot{U}} (ξ ξ^{A} \dot{η} \dot{U} \end{aligned} η^{A} {\bar{ξ}}^{\dot{U}}) = 0, $ \end{matrix}

and spacelike,

\begin{aligned} v \cdot y & = - \frac{1}{2} (ξ_{A} {\bar{η}}_{\dot{U}} + η_{A} {\bar{ξ}}_{\dot{U}}) (ξ^{A} {\bar{η}}^{\dot{U}} + η^{A} {\bar{ξ}}^{\dot{U}}) \\ (41.94) & = - \frac{1}{2} (ξ_{A} η^{A}) ({\bar{η}}_{\dot{U}} {\bar{ξ}}^{\dot{U}}) - \frac{1}{2} (η_{A} ξ^{A}) ({\bar{ξ}}_{\dot{U}} {\bar{η}}^{\dot{U}}) = 1 \end{aligned}

("unit length of flag").
Multiplication of the spinor

ξ^{A}

by the "flag factor"

e^{i σ}

rotates the flag about the flagpole by the angle

2 σ

, because the spinor mate,

η^{A}

, of

ξ^{A}

is multiplied by the factor

e^{- i σ}

[see the normalization condition (41.81)]. These changes alter the vector

y

to a rotated vector

y_{rot}

, with

\begin{aligned} y_{rot}^{α} ⟶ Y_{rot}^{A \dot{O}} = & e^{2 i σ ξ^{A} {\bar{η}}^{\dot{U}} + e^{- 2 i σ} η^{A} {\bar{ξ}}^{\dot{U}}} \\ = & \cos 2 σ (ξ^{A} {\bar{η}}^{\dot{U}} + η^{A} {\bar{ξ}}^{\dot{U}}) + \sin 2 σ (i ξ^{A} {\bar{η}}^{\dot{U}} - i η^{A} {\bar{ξ}}^{\dot{U}}) \\ (41.95) & ⟶ y^{α} \cos 2 σ + z^{α} \sin 2 σ . \end{aligned}

Here the 4 -vector

z

shares with the vector

y

these properties: it is (1) real, (2) spacelike, (3) of unit magnitude, (4) orthogonal to the null 4 -vector

x

of the flagpole, and (5) uniquely specified by the original spinor

ξ^{A}

up to the additive real

Rotation of flag about flagpole

Equations relating spinor, flagpole, and flag
multiple

(α + \bar{α})

x

. In addition,

z

and

y

are orthogonal. Thus

y

and

z

provide basis vectors in the two-dimensional space in which-to overpictorialize-the "tip of the flag" undergoes its rotation.

Recapitulate by returning to the laser pulse. Two numbers, such as the familiar polar angles

θ

(angle with the

z

-axis) and

ϕ

(azimuth around

z

-axis from

x

-axis) tell the direction of its flight. A third number,

r

, gives the distance to the moon and also the travel time for light to reach the moon. A fourth number, an angle

ψ

, tells the azimuth of the illuminated arrow shot onto the surface of the moon, this azimuth to be measured from the

e_{θ}

direction (where

ψ = 0

), around the flagpole in a righthanded sense. Then the spinor associated with the flagpole plus flag (rotated arrow) is

\begin{matrix} (41.96) & (\binom{ξ^{1}}{ξ^{2}}) = (2 r)^{1 / 2} (\binom{\cos (θ / 2) e^{- i ϕ / 2 + i ψ / 2}}{\sin (θ / 2) e^{i ϕ / 2 + i ψ / 2}}) \end{matrix}

according to the conventions adopted here [see (41.87)]. The mate

η_{A}

to this spinor, unique up to an additive multiple of

ξ^{A}

, is

\begin{matrix} (41.97) & (\binom{η^{1}}{η^{2}}) = (2 r)^{- 1 / 2} (\binom{- \sin (θ / 2) e^{- i ϕ / 2 - i ψ / 2}}{\cos (θ / 2) e^{i ϕ / 2 - i ψ / 2}}) . \end{matrix}

The 4 -vector of the flagpole determined by

ξ^{4}

is found from (41.87):

\begin{matrix} (41.98) & (\begin{array}{l} x^{0} \\ x^{1} \\ x^{2} \\ x^{3} \end{array}) = (\begin{matrix} r \\ r \sin θ \cos ϕ \\ r \sin θ \sin ϕ \\ r \cos θ \end{matrix}) . \end{matrix}

To determine the flag itself, one requires, in addition to

x^{α}

, the unit spacelike 4-vector

y^{α}

, normal to

x^{α}

, and unique up to an additive real multiple of

x^{α}

. This vector is evaluated by use of (41.90) and has the form

\begin{matrix} (41.99) & (\begin{matrix} y^{0} \\ y^{1} \\ y^{2} \\ y^{3} \end{matrix}) = (\begin{matrix} 0 \\ \cos θ \cos ϕ \cos ψ + \sin ϕ \sin ψ \\ \cos θ \sin ϕ \cos ψ - \cos ϕ \sin ψ \\ - \sin θ \cos ψ \end{matrix}) . \end{matrix}

From these expressions for

x^{μ}

and

y^{μ}

, one calculates the components of the bivector ("flag")

F^{μ ν} = x^{μ} y^{ν} - y^{μ} x^{ν}

by simple arithmetic.

§41.10. APPEARANCE OF THE NIGHT SKY: AN APPLICATION OF SPINORS

Attention has gone here to extracting all four pieces of information contained in a spinor: separation in time (equal to separation in space), direction in space, and

Figure 41.8.
Representation of a direction in space (one of the stars of the Big Dipper, regarded as a point on the celestial sphere) as a point in the complex

ζ

plane

(ζ =

ratio

ξ^{2} / ξ^{1}

of spinor components) by stereographic projection from the South Pole.
rotation about that direction. Turn now to an application where not all that information is needed. Look at the night sky and ask (1) how to describe its appearance and (2) how to change that appearance. As one way to describe its appearance, give the direction of each star. Abandon any concern about the distance of the star, and any concern about any rotation

ψ

about the flagpole. In other words, the complex factor

(2 r)^{1 / 2} e^{i ψ / 2}

common to

ξ^{1}

and

ξ^{2}

drops from attention. All that is left as significant is the ratio

ζ

of these spinor components:

\begin{matrix} (41.100) & ζ = ξ^{2} / ξ^{1} = \tan (θ / 2) e^{i ϕ} . \end{matrix}

To give the one complex number

ζ

("stereographic coordinate;" Figure 41.8) for each star in the sky is to catalog the pattern of the stars.

Let the observer change his stance. The celestial sphere appears to rotate. Or let him rocket past his present location in the direction of the North Star with some substantial fraction of the velocity of light. To him all that portion of the celestial sphere is opened out, as if by a magnifying glass. To compensate, the remaining stars are packed into a smaller angular compass. Any such rotation or boost or combination of rotation and boost being described in spinor language by a transformation of the form

\begin{matrix} (41.101) & ξ^{A} ⟶ ξ_{new}^{A} = L_{B}^{A} ξ_{B}^{B}, \end{matrix}

Spinors used to analyze "Lorentz transformations" of appearance of night sky
implies a transformation of the complex stereographic coordinate of any given star of the form

\begin{matrix} (41.102) & ζ ⟶ ζ_{new} = \frac{ξ_{new}^{2}}{ξ_{new}^{1}} = \frac{L_{2}^{2} ζ + L^{2}}{L_{2}^{1} ζ + L_{1}^{1}} \end{matrix}

In the special case of a boost in the

z

-direction with velocity parameter

α

(velocity

β = \tanh α

), the off-diagonal components

L^{1}_{2}

and

L^{2}_{1}

vanish. The magnification of the overhead sky then expresses itself in the simple formula

ζ_{new} = e^{α} ζ

\begin{matrix} ϕ_{new} = ϕ \\ (41.103) & \tan (θ_{new} / 2) = e^{α} \tan (θ / 2) \end{matrix}

Contrary to this prediction and false expectation, no magnification at all is achieved of the regions around the North Star by moving with high velocity in that direction. On the contrary, any photon coming in from a star a little off that direction, with a little transverse momentum, keeps that transverse momentum in the new frame; but its longitudinal momentum against the observer is augmented by his motion. Thus the ratio of the momenta is decreased, and the observed angle relative to the North Star is also decreased. The consequence is not magnification, but diminution ("looking through the wrong end of a telescope"). The correct formula is not (41.103) but

\begin{matrix} (41.104) & \tan (θ_{new} / 2) = e^{- α} \tan (θ / 2) \end{matrix}

(reversal of the sign of

α

). The reason for this correction is not far to seek. The spinor analysis so far had dealt with an outgoing light pulse, and a 4 -vector with positive time component. That feature was built into the formula adopted to tie the spinor to the 4 -vector,

\begin{matrix} (41.105) & r 1 + (r \cdot σ) \equiv X = ‖ ξ^{A} {\bar{ξ}}^{\dot{U}} ‖ . \end{matrix}

In contrast the 4 -vector that reaches back to the origin of an incoming photon has a time component that is negative (or, alternatively, sign-reversed space components)! For any null 4 -vector with negative time component, one employs instead of (41.105) the formula

\begin{matrix} (41.106) & X = - ‖ ξ ξ^{A} \bar{ξ} {\dot{U}}_{‖} . \end{matrix}

It is enough to mention here this point of principle without going through the details that give the altered sign for

α

in (41.104). From now on, to preserve the previous arithmetic, change the problem. Deal, not with incoming photons, but with outgoing photons. Replace the receiving telescope by the projector of a planetarium. It projects out into space a separate beam of light for each star of the Big Dipper and also one for the North Star itself. Let an observer move in the positive

z

-direction with velocity parameter

α

. In his frame of reference the beams actually will be widened out in full accord with (41.103).
"The magnification process changes the size of the Big Dipper but not its shape."

This statement is at the same time true and false. It is true of the Dipper and of any other constellation to the extent that the angular dimensions of that constellation can be idealized to be small compared to the entire compass of the sky. It is false in the sense that any well-rounded projected constellation, however small it may appear to an observer at rest with respect to the earth, can always be so "opened out" by the observer putting on any sufficiently high velocity, the observer still being near the earth, that the constellation encompasses a major fraction of the sky.

That the "Lorentz-transformation-induced magnification" of a small object does not change its shape can be seen in three ways. (1) Stereographic projection (Figure 41.8) and "fractional linear transformation" (41.102) are both known to leave all angles unchanged ["conformal invariance;" see for example Penrose (1959)] and known even to turn every old circle into a new circle. (2) Consider a given star,

M

, in the constellation and immediate neighbors,

L

and

N

, just below it and just above it in the count of the members of that constellation. Consider the flagpole pointed at

M

and the flag pointed first from

M

L

, then from

M

N

. The flag has turned about the flagpole through an angle

ψ

. The two corresponding spinors therefore differ by a phase factor

e^{i ψ / 2}

. They differ in no other way. After an arbitrary Lorentz transformation they still differ by the phase factor

e^{i ψ / 2}

, and in no other way. The angle between the arcs

M L

and

M N

on the celestial sphere therefore remains at its original value

ψ

after the Lorentz transformation (again conformal invariance of patterns on the celestial sphere!). (3) An even more elementary calculation shows that infinitesimal arc lengths on the unit celestial sphere in the direction of increasing

θ

and arc lengths in the direction of increasing

ϕ

are magnified in the same proportion, thus leaving unchanged the angle between arc and arc (conformal invariance). Thus, consider a photon shot out from the planetarium projector to a point on the celestial sphere ("planetarium version of a Big-Dipper star") with inclination

θ

to the

z

-axis, as seen by an observer at rest relative to the earth. From the standard laws of transformation of angles in a Lorentz transformation ("aberration"; Box 2.4), one has for the sine of the transformed angle

\begin{matrix} (41.107) & \sin θ_{new} = \frac{{(1 - β^{2})}^{1 / 2}}{1 - β \cos θ} \sin θ \end{matrix}

and (by differentiating the expression for the cosine of the transformed angle)

\begin{matrix} (41.108) & d θ_{new} = \frac{{(1 - β^{2})}^{1 / 2}}{1 - β \cos θ} d θ \end{matrix}

From these expressions it follows at once that the inclination, relative to a meridian line, on the transformed celestial sphere is identical to the direction, relative to the same meridian line, on the original celestial sphere:

\begin{aligned} \tan (\binom{new}{inclination}) & = \frac{\sin θ_{new} d ϕ_{new}}{d θ_{new}} = \frac{\sin θ d ϕ}{d θ} \\ (41.109) & = \tan (\binom{original}{inclination}) \end{aligned}

Lorentz transformations leave angles on sky unchanged ("conformal invariance")

So much for the elementary spinor and what it has to do with a null vector, with a "flagpole" pointed to the celestial sphere, and with rotation of a "flag" about such a flagpole.

§41.11. SPINORS AS A POWERFUL TOOL IN GRAVITATION THEORY

Spinor formalism in curved spacetime

Spinors needed when analyzing fermions in gravitational fields

Equivalence of spinor and tensor formalisms

Just as vectors, tensors, and differential forms are easily generalized from flat spacetime to curved, so are spinors.

Each event

P

in curved spacetime possesses a tangent space. In that tangent space reside and operate all the vectors, tensors, and forms located at

P

. The geometry of the tangent space is Lorentzian ("local Lorentz geometry at

P

"), since the scalar product of any two vectors

u

and

v

P

, expressed in an orthonormal frame at

P

, is

u \cdot v = g (u, v) = η_{\hat{α} \hat{β}} u^{\hat{α}} v^{\hat{β}}

Thus, there is no mathematical difference between the tangent space at

P

on the one hand, and flat spacetime on the other. Whatever mathematical can be done in the one can also be done in the other. In particular, the entire formalism of spinors, developed originally in flat spacetime, can be carried over without change to the tangent space at the arbitrary event

P

in curved spacetime.

Let it be done. Now spinors reside at every event in curved spacetime; and at each event one can translate back and forth between spinor language and tensor language, using the equations (valid in orthonormal frames) of

§ § 41.6

and 41.7.

Spinors in curved spacetime are an indispensible mathematical tool, when one wishes to study the influence of gravity on quantized particles of half-integral spin (neutrinos, electrons, protons, . . .). Consider, for example, Hartle's (1971) proof that a black hole cannot exert any long-range, weak-interaction forces on external matter (i.e., that a black hole has no "weak-interaction hair"). His proof could not function without a spinor description of neutrino fields in curved spacetime. Similarly for Wheeler's (1971b) analysis of the quasibound states of an electron in the gravitational field of a small black hole (gravitational radius

\sim 10^{- 13} cm

): it requires solving the Dirac equation for a spin-

\frac{1}{2}

particle in the curved spacetime geometry of Schwarzschild. For a detailed discussion of the Dirac equation in curved spacetime see, e.g., Brill and Wheeler (1957).

To use the mathematics of spinors, one need not be dealing with quantum theory or with particles of half-integral spin. The spinor formalism is perfectly applicable in situations where only integral-spin entities (scalars, vectors, tensors) are in view, and where in fact, the spinor formalism is fully equivalent to the tensor formalism that pervades earlier chapters of this book. Equations (41.77) and (41.78) provide the translation from one formalism to the other, once an orthonormal frame has been chosen at each event in spacetime.

Certain types of problems in gravitation theory are far more tractable in the language of spinors than in the language of tensors. Examples are as follows.

(1) Geometric Optics (the theory of "null congruences of geodesics")

Here spinors make almost trivial the lengthy tensor algebra needed in derivations of the "focusing theorem" [equation (22.37)]; and they yield an elegant, simple formalism for discussing and calculating how, with increasing affine parameter, a bundle of rays alters its size ("expansion"), its shape ("shear"), and its orientation ("rotation"). See, e.g., Sachs (1964), Pirani (1965), or Penrose (1968a) for a review and the original references.

(2) Radiation Theory in Curved Spacetime (both gravitational and electromagnetic)

Spinors provide the most powerful of all formalisms for decomposing radiation fields into spherical harmonics and for manipulating their decomposed components. See, for example, Price's (1972a,b) analysis of how a perturbed Schwarzschild black hole radiates away all its radiatable perturbations, be they electromagnetic perturbations, gravitational perturbations, or perturbations in a fictitious field of spin 17; see, similarly, the analysis by Fackerell and Ipser (1972) and by Ipser (1971) of electromagnetic perturbations of a Kerr black hole, and the analysis by Teukolsky (1972) of gravitational perturbations of a Kerr hole. Spinors also yield an elegant and powerful analysis of the "

1 / r

" expansion of a radiation field flowing out from a source into asymptotically flat space. Among its results is a "peeling theorem," which describes the algebraic properties of the coefficients in a

1 / r

expansion of the Riemann tensor. See, e.g., Sachs (1964) or Pirani (1965) for reviews and original references.

(3) Algebraic Properties of Curvature Tensors

The spinor formalism is a more powerful method than any other for deriving the "Petrov-Pirani algebraic classification of the conformal curvature tensor," and for proving theorems about algebraic properties of curvature tensors. See, e.g., Sachs (1964) or Pirani (1965) or Penrose (1968a) for reviews and references.

Of course, the spinor formalism, like any formalism, has its limitations. For example, many of the elementary problems of gravitation theory, and a large fraction of the most difficult ones, would be more difficult in the language of spinors than in the language of tensors! But for certain classes of problems, especially those where null vectors play a central role, spinors are a most valuable tool.

Cartan gave spinors to the world's physics and mathematics. His text (American edition, 1966) is an important reference to the subject.

Applications of spinor formalism in classical gravitation theory

синатен 42

REGGE CALCULUS

This chapter is entirely Track 2. As preparation for it, Chapter 21 (variational principle and initial-value formalism) is needed. It is not needed as preparation for any later chapter, though it will be helpful in Chapter 43 (dynamics of geometry).

The need for Regge calculus as a computational tool

§42.1. WHY THE REGGE CALCULUS?

Gravitation theory is entering an era when situations of greater and greater complexity must be analyzed. Before about 1965 the problems of central interest could mostly be handled by idealizations of special symmetry or special simplicity or both. The Schwarzschild geometry and its generalizations, the Friedmann cosmology and its generalizations, the joining together of the Schwarzschild geometry and the Friedmann geometry to describe the collapse of a bounded collection of matter, the vibrations of relativistic stars, weak gravitational waves propagating in an otherwise flat space: all these problems and others were solved by elementary means.
But today one is pressed to understand situations devoid of symmetry and not amenable to perturbation theory: How do two black holes alter in shape, and how much gravitational radiation do they emit when they collide and coalesce? What are the structures and properties of the singularities at the endpoint of gravitational collapse, predicted by the theorems of Penrose, Hawking, and Geroch? Can a Universe that begins completely chaotic smooth itself out quickly by processes such as inhomogeneous mixmaster oscillations?
To solve such problems, one needs new kinds of mathematical tools-and in response to this need, new tools are being developed. The "global methods" of Chapter 34 provide one set of tools. The Regge Calculus provides another [Regge (1961); see also pp. 467-500 of Wheeler (1964a)].

§42.2. REGGE CALCULUS IN BRIEF

Consider the geodesic dome that covers a great auditorium, made of a multitude of flat triangles joined edge to edge and vertex to vertex. Similarly envisage space-

Approximation of smooth geometries by skeleton structures
time, in the Regge calculus, as made of flat-space "simplexes" (four-dimensional
item in this progression: two dimensions, triangle; three dimensions, tetrahedron; four dimensions, simplex) joined face to face, edge to edge, and vertex to vertex. To specify the lengths of the edges is to give the engineer all he needs in order to know the shape of the roof, and the scientist all he needs in order to know the geometry of the spacetime under consideration. A smooth auditorium roof can be approximated arbitrarily closely by a geodesic dome constructed of sufficiently small triangles. A smooth spacetime manifold can be approximated arbitrarily closely by a locked-together assembly of sufficiently small simplexes. Thus the Regge calculus, reaching beyond ordinary algebraic expressions for the metric, provides a way to analyze physical situations deprived, as so many situations are, of spherical symmetry, and systems even altogether lacking in symmetry.

If the designer can give the roof any shape he pleases, he has more freedom than the analyst who is charting out the geometry of spacetime. Given the geometry of spacetime up to some spacelike slice that, for want of a better name, one may call "now," one has no freedom at all in the geometry from that instant on. Einstein's geometrodynamic law is fully deterministic. Translated into the language of the Regge calculus, it provides a means to calculate the edge lengths of new simplexes from the dimensions of the simplexes that have gone before. Though the geometry is deterministically specified, how it will be approximated is not. The original spacelike hypersurface ("now") is approximated as a collection of tetrahedrons joined together face to face; but how many tetrahedrons there will be and where their vertices will be placed is the option of the analyst. He can endow the skeleton more densely with bones in a region of high curvature than in a region of low curvature to get the most "accuracy profit" from a specified number of points. Some of this freedom of choice for the lengths of the bones remains as one applies the geometrodynamic law in the form given by Regge (1961) to calculate the future from the past. This freedom would be disastrous to any computer program that one tried to write, unless the programmer removed all indefiniteness by adding supplementary conditions of his own choice, either tailored to give good "accuracy profit," or otherwise fixed.

Having determined the lengths of all the bones in the portion of skeletonized spacetime of interest, one can examine any chosen local cluster of bones in and by themselves. In this way one can find out all there is to be learned about the geometry in that region. Of course, the accuracy of one's findings will depend on the fineness with which the skeletonization has been carried out. But in principle that is no limit to the fineness, or therefore to the accuracy, so long as one is working in the context of classical physics. Thus one ends up with a catalog of all the bones, showing the lengths of each. Then one can examine the geometry of whatever spacelike surface one pleases, and look into many other questions besides. For this purpose one has only to pick out the relevant bones and see how they fit together.

§42.3. SIMPLEXES AND DEFICIT ANGLES

Figure 42.1 recalls how a smoothly curved surface can be approximated by flat triangles. All the curvature is concentrated at the vertices. No curvature resides at

Role of Einstein field equation in fixing the skeleton structure

Figure 42.1.
A 2-geometry with continuously varying curvature can be approximated arbitrarily closely by a polyhedron built of triangles, provided only that the number of triangles is made sufficiently great and the size of each sufficiently small. The geometry in each triangle is Euclidean. The curvature of the surface shows up in the amount of deficit angle at each vertex (portion

A B C D

of polyhedron laid out above on a flat surface).

Deficit angle as a skeletonized measure of curvature:
(1) in two dimensions
(2) in

n

(or four) dimensions
the edge between one triangle and the next, despite one's first impression. A vector carried by parallel transport from

A

through

B

and

C

D

, and then carried back by another route through

C

and

B

A

returns to its starting point unchanged in direction, as one sees most easily by laying out this complex of triangles on a flat surface. Only if the route is allowed to encircle the vertex common to

A, B, C

, and

D

does the vector experience a net rotation. The magnitude of the rotation is equal to the indicated deficit angle,

δ

, at the vertex. The sum of the deficit angles over all the vertices has the same value,

4 π

, as does the half-integral of the continuously distributed scalar curvature

(^{(2)} R = 2 / a^{2}

for a sphere of radius

a

) taken over the entirety of the original smooth figure,

\begin{matrix} (42.1) & \sum_{\begin{matrix} skeleton \\ geometry \end{matrix}} δ_{i} = \frac{1}{2} \int_{\begin{matrix} actual smooth \\ geometry \end{matrix}}^{(2)} R d (surface) = 4 π . \end{matrix}

Generalizing from the example of a 2-geometry, Regge calculus approximates a smoothly curved

n

-dimensional Riemannian manifold as a collection of

n

-dimensional blocks, each free of any curvature at all, joined by

(n - 2)

-dimensional regions in which all the curvature is concentrated (Box 42.1). For the four-dimensional spacetime of general relativity, the "hinge" at which the curvature is concentrated has the shape of a triangle, as indicated schematically in the bottom row of Figure 42.2. In the example illustrated there, ten tetrahedrons have that triangle in common. Between one of these tetrahedrons and the next fits a four-dimensional simplex. Every feature of this simplex is determined by the lengths of its ten edges. One of the features is the angle

α

between one of the indicated tetrahedrons or "faces" of the simplex and the next. Thus

α

represents the angle subtended by this simplex

Box 42.1 THE HINGES WHERE THE CURVATURE IS CONCENTRATED IN THE "ANGLE OF RATTLE" BETWEEN BUILDING BLOCKS IN A SKELETON MANIFOLD

Dimensionality of manifold

Elementary flat-space

building block:

Edge lengths to define it:

triangle

tetrahedron

simplex

Hinge where cycle of such

blocks meet with a deficit

angle or "angle of rattle"

δ :

Dimensionality of hinge:

"Content" of such a hinge:

Contribution from all hinges

within a given small region

to curvature of manifold:

vertex

edge

Continuum limit of this quantity
expressed as an integral over the same small region:

\frac{1}{2} \int^{(2)} R {(^{(2)} g)}^{1 / 2} d^{2} x \frac{1}{2} \int^{(3)} R {(^{(3)} g)}^{1 / 2} d^{3} x \frac{1}{2} \int^{(4)} R {(-^{(4)} g)}^{1 / 2} d^{4} x

at the hinge. Summing the angles

α

for all the simplexes that meet on the given hinge

P R

, and subtracting from

2 π

, one gets the deficit angle associated with that hinge. And by then summing the deficit angles in a given small

n

-volume with appropriate weighting (Box 42.1), one obtains a number equal to the volume integral of the scalar curvature of the original smooth

n

-geometry. See Box 42.2.

§42.4. SKELETON FORM OF FIELD EQUATIONS

Rather than translate Einstein's field equations directly into the language of the skeleton calculus, Regge turns to a standard variational principle from which Einstein's law lets itself be derived. It says (see

$ $ 21.2

and 43.3 ) adjust the 4 -geometry throughout an extended region of spacetime, subject to certain specified conditions on the boundary, so that the dimensionless integral (action in units of

ℏ!

\begin{matrix} (42.2) & I = (c^{3} / 16 π ℏ G) \int R (- g)^{1 / 2} d^{4} x \end{matrix}

is an extremum. This statement applies when space is free of matter and electromag-

Einstein-Hilbert variational principle reduced to skeleton form

Figure 42.2.
Cycle of building blocks associated with a single hinge. Top row, two dimensions: left, schematic association of vertices

S, T, Q, T, W

with "hinge" at the vertex

P

; right, same, but with elementary triangles indicated in full. Middle row, three dimensions: left, schematic; right, perspective representation of the six tetrahedrons that meet on the "hinge"

P

. Bottom row, four dimensions; shown only

the interior of which space is flat. The five vertices

P R Q E

belong to the next simplex; and so on around the cycle of simplexes. The two simplexes just named interface at the tetrahedron

P Q Q Q

, inside which

angle

α

subtended at the "hinge"

P Q R

. The value of this angle is completely fixed by the ten edge

subtended at the hinge

P Q R

by the other simplexes of the cycle, do not in general add up to

2 π

. The deficit, the "angle of rattle" or deficit angle

δ

, gives the amount of curvature concentrated at the hinge

P

QR. There is no actual rattle or looseness of fit, unless one tries to imbed the cycle into an over-all flat four-dimensional space (analog of "stamping on" the collection of triangles, and seeing them open out by the amount of the deficit angle, as indicated in inset in Figure 42.1).
netic fields, a simplification that will be made in the subsequent discussion to keep it from becoming too extended. When in addition all lengths are expressed in units of the Planck length

\begin{matrix} (42.3) & L^{*} = {(ℏ G / c^{3})}^{1 / 2} = 1.6 \times 10^{- 33} cm, \end{matrix}

and the curvature integral is approximated by its expression in terms of deficit angles, Regge shows that the statement

δ I = 0

(condition for an extremum!) becomes

\begin{matrix} (42.4) & (1 / 8 π) δ \sum_{\begin{matrix} hinge. \\ h = 1 \end{matrix}}^{H} A_{h} δ_{h} = 0 . \end{matrix}

Box 42.2 FLOW DIAGRAMS FOR REGGE CALCULUS

A skeleton 4-geometry is completely determined by all its edge lengths. From the edge lengths one gets the integrated curvature by pursuing, for each hinge in the 4-geometry, the following flow diagram:

One finds it natural to apply this analysis in either of two ways. First, one can probe a given 4 -geometry (given set of edge lengths!) in the sense

Box 42.2 (continued)

Second-and this is the rationale of Regge calculus-one can use the skeleton calculus to deduce a previously unknown 4-geometry from Einstein's geometrodynamic law, proceeding in the direction

In the changes contemplated in this variational principle, certain edge lengths are thought of as being fixed. They have to do with the conditions specified at the boundaries of the region of spacetime under study. It is not necessary here to enter into the precise formulation of these boundary conditions, fortunately, since some questions of principle still remain to be clarified about the precise formulation of boundary conditions in general relativity (see §21.12). Rather, what is important is the effect of changes in the lengths of the edges of the blocks in the interior of the region being analyzed, as they augment or decrease the deficit angles at the various hinges. In his basic paper on the subject, Regge (1961) notes that the typical deficit angle

δ_{h}

depends in a complicated trigonometric way on the values of numerous edge lengths

ℓ_{p}

. However, he proves (Appendix of his paper) that "quite remarkably, we can carry out the variation as if the

δ_{h}

were constants," thus reducing the variational principle to the form

\begin{matrix} (42.5) & (1 / 8 π) \sum_{\begin{matrix} hinges \\ h i = 1 \end{matrix}}^{H} δ_{h} δ A_{h} = 0 . \end{matrix}

Here the change in area of the

h

-th triangle-shaped hinge, according to elementary trigonometry, is

\begin{matrix} (42.6) & δ A_{h} = \frac{1}{2} \sum_{p} ℓ_{p} δ ℓ_{p} cotan θ_{p h} . \end{matrix}

In this equation

θ_{p h}

is the angle opposite to the

p

-th edge in the triangle. Consequently, Einstein's equations in empty space reduce in skeleton geometry to the form

\begin{matrix} (42.7) & \sum_{\begin{matrix} hinges that \\ have the \\ give edge \\ p in common \end{matrix}} δ_{h} cotan θ_{p h} = 0, (p = 1, 2, \dots), \end{matrix}

one equation for each edge length in the interior of the region of spacetime being analyzed.

§42.5. THE CHOICE OF LATTICE STRUCTURE

Two questions arise in the actual application of Regge calculus, and it is not clear that either has yet received the resolution which is most convenient for practical applications of this skeleton analysis: What kind of lattice to use? How best to capitalize on the freedom that exists in the choice of edge lengths? The first question is discussed in this section, the second in the next section.

It might seem most natural to use a lattice made of small, nearly rectangular blocks, the departure of each from rectangularity being conditioned by the amount and directionality of the local curvature. However, such building blocks are "floppy." One could give them rigidity by specifying certain angles as well as the edge lengths. But then one would lose the cleanness of Regge's prescription: give edge lengths, and give only edge lengths, and give each edge length freely and independently, in order to define a geometry. In addition one would have to rederive the Regge equations, including new equations for the determination of the new angles. Therefore one discards the quasirectangle in favor of the simplex with its

5 \cdot 4 / 2 = 10

edge lengths. This decided, one also concludes that even in flat spacetime the simplexes cannot all have identical edge lengths. Two-dimensional flat space can be filled with identical equilateral triangles, but already at three dimensions it ceases to be possible to fill out the manifold with identical equilateral tetrahedrons. One knows that a given carbon atom in diamond is joined to its nearest neighbors with tetrahedral bonds, but a little reflection shows that the cell assignable to the given atom is far from having the shape of an equilateral tetrahedron.

Synthesis would appear to be a natural way to put together the building blocks: first make one-dimensional structures; assemble these into two-dimensional structures; these, into three-dimensional ones; and these, into the final four-dimensional structure. The one-dimensional structure is made of points,

1, 2, 3, \dots

, alternating with line segments,

12, 23, 34, \dots

. To start building a two-dimensional structure, pick up a second one-dimensional structure. It might seem natural to label its points

1^{'}, 2^{'}, 3^{'}, \dots

, etc. However, that labeling would imply a cross-connection between 1 and

1^{'}

, between 2 and

2^{'}

, etc., after the fashion of a ladder. Then the elementary cells would be quasirectangles. They would have the "floppiness" that is to be excluded. Therefore relabel the points of the second one-dimensional structure as

1 {\frac{1}{2}}^{\frac{1}{2}}, 2 {\frac{1}{2}}^{'}, 3 {\frac{1}{2}}^{'}

, etc. The implication is that one cross-connects

2 \frac{1^{'}}{}

with points 2 and 3 of the original one-dimensional structure, etc. One ends up with something like the

Einstein field equation reduced to skeleton form

The choice of lattice structure:
(1) avoiding floppiness
(2) necessity for unequal edge lengths
(3) construction of twodimensional structures

(4) 3-D structures built from 2-D structures by "method of blocks"
(5) 3-D structures from 2-D by "method of spheres"
girder structure of a bridge, fully rigid in the context of two dimensions, as desired. The same construction, extended, fills out the plane with triangles. One now has a simple, standard two-dimensional structure. One might mistakenly conclude that one is ready to go ahead to build up a three-dimensional structure: the mistake lies in the tacit assumption that the flat-space topology is necessarily correct.

Let it be the problem, for example, to determine the development in time of a 3 -geometry that has the topology of a 3 -sphere. This 3 -sphere is perhaps strongly deformed from ideality by long-wavelength gravitational waves. A right arrangement of the points is the immediate desideratum. Therefore put aside for the present any consideration of the deformation of the geometry by the waves (alteration of edge lengths from ideality). Ask how to divide a perfect 3-sphere into two-dimensional sheets. Here each sheet is understood to be separated from the next by a certain distance. At this point two alternative approaches suggest themselves that one can call for brevity "blocks" and "spheres."
(1) Blocks. Note that a 3-sphere lets itself be decomposed into 5 identical, tetra-hedron-like solid blocks ( 5 vertices; 5 ways to leave out any one of these vertices!) Fix on one of these "tetrahedrons." Select one vertex as summit and the face through the other three vertices as base. Give that base the two-dimensional lattice structure already described. Introduce a multitude of additional sheets piled above it as evenly spaced layers reaching to the summit. Each layer has fewer points than the layer before. The decomposition of the 3-geometry inside one "tetrahedron" is thereby accomplished. However, an unresolved question remains; not merely how to join on this layered structure in a regular way to the corresponding structure in the adjacent "tetrahedrons"; but even whether such a regular joinup is at all possible. The same question can be asked about the other two ways to break up the 3-sphere into identical "tetrahedrons" [Coxeter (1948), esp. pp. 292-293: 16 tetrahedrons defined by a total of 8 vertices or 600 tetrahedrons defined by a total of 120 vertices]. One can eliminate the question of joinup of structure in a simple way, but at the price of putting a ceiling on the accuracy attainable: take the stated number of vertices ( 5 or 8 or 120) as the total number of points that will be employed in the skeletonization of the 3 -geometry (no further subdivision required or admitted). Considering the boundedness of the memory capacity of any computer, it is hardly ridiculous to contemplate a limitation to 120 tracer points in exploratory calculations!
(2) Spheres. An alternative approach to the "atomization" of the 3-sphere begins by introducing on the 3 -sphere a North Pole and a South Pole and the hyperspherical angle

χ (χ = 0

at the first pole,

χ = π

at the second,

χ = π / 2

at the equator; see Box 27.2). Let each two-dimensional layer lie on a surface of constant

χ

(

χ

equal to some integer times some interval

Δ χ

). The structure of this 2 -sphere is already to be regarded as skeletonized into elementary triangles ("fully complete Buckminster Fuller geodesic dome"). Therefore the number of "faces" or triangles

F

, the number of edge lengths

E

, and the number of vertices

V

must be connected by the relation of Euler:

\begin{matrix} (42.8) & F - E + V = (\binom{a topology-dependent}{number or "Euler character"}) = {\begin{cases} 2 for 2-sphere, \\ 0 for 2-torus. \end{cases} \end{matrix}

It follows from this relation that it is impossible for each vertex to sit at the center
of a hexagon (each vertex the point of convergence of 6 triangles). This being the case, one is not astonished that a close inspection of the pattern of a geodesic dome shows several vertices where only 5 triangles meet. It is enough to have 12 such 5 -triangle vertices among what are otherwise all 6 -triangle vertices in order to meet the requirements of the Euler relation:

\begin{aligned} n & 5-triangle vertices \\ V - n & 6-triangle vertices \\ F & = (V - n) (6 / 3) + n (5 / 3) triangles \\ (42.9) & E & = (V - n) (6 / 2) + n (5 / 2) edges \\ V & = (V - n) (6 / 6) + n vertices \\ 2 = F - E + V & = n / 6 Euler characteristic \\ n & = 12 \end{aligned}

Among all figures with triangular faces, the icosahedron is the one with the smallest number of faces that meets this condition (5-triangle vertices exclusively!)

If each 2 -surface has the pattern of vertices of a geodesic dome, how is one dome to be joined to the next to make a rigid skeleton 3-geometry? Were the domes imbedded in a flat 3-geometry, rigidity would be no issue. Each dome would already be rigid in and by itself. However, the 3 -geometry is not given to be flat. Only by a completely deterministic skeletonization of the space between the two 2 -spheres will they be given rigidity in the context of curved space geometry. (1) Not by running a single connector from each vertex in one surface to the corresponding vertex in the next ("floppy structure"!) (2) Not by displacing one surface so each of its vertices comes above, or nearly above, the center of a triangle in the surface "below." First, the numbers of vertices and triangles ordinarily will not agree. Second, even when they do, it will not give the structure the necessary rigidity to connect the vertex of the surface above to the three vertices of the triangle below. The space between will contain some tetrahedrons, but it will not be throughout decomposed into tetrahedrons. (3) A natural and workable approach to the skeletonization of the 3 -geometry is to run a connector from each vertex in the one surface to the corresponding vertex in the next, but to flesh out this connection with additional structure that will give rigidity to the 3 -geometry: intervening vertices and connectors as illustrated in Box 42.3.

In working up from the skeletonization of a 3-geometry to the skeletonization of a 4-geometry, it is natural to proceed similarly. (1) Use identical patterns of points in the two 3 -geometries. (2) Tie corresponding points together by single connectors. (3) Halfway, or approximately half way between the two 3-geometries insert a whole additional pattern of vertices. Each of these supplementary vertices is "dual" to and lies nearly "below" the center of a tetrahedron in the 3-geometry immediately above. (4) Connect each supplementary vertex to the vertices of the tetrahedron immediately above, to the vertices of the tetrahedron immediately below, and to those other supplementary vertices that are its immediate neighbors. (5) In this way get the edge lengths needed to divide the 4 -geometry into simplexes, each of rigidly defined dimensions.
(6) 4-D structures built from 3-D structures

Box 42.3 SYNTHESIS OF HIGHER-DIMENSIONAL SKELETON GEOMETRIES OUT OF LOWER-DIMENSIONAL SKELETON GEOMETRIES

(1) One-dimensional structure as alternation of points and line segments. (2) Two-dimensional structure (a) "floppy" (unacceptable) and (b) rigidified (angles of triangles fully determined by edge lengths). When this structure is extended, as at right, the "normal" vertex has six triangles hinging on it. However, at least twelve 5 -triangle vertices of the type indicated at

a

are to be interpolated if the 2 -geometry is to be able to close up into a 2 -sphere. (3) Skeleton 3 -geometry obtained by filling in between the skeleton 2 -geometry...

Q B \dots F P C \dots E D \dots

and the similar structure

\dots a^{'} B^{'} \dots F^{'} P^{'} C^{'} \dots E^{'} D^{'} \dots

as follows. (a) Insert direct connectors such as

P P^{'}

between corresponding points in the two 2 -geometries. (b) Insert an intermediate layer of "supplementary vertices" such as

5 T V T F .

. . Each of these supplementary vertices lies roughly halfway between the center of the triangle "above" it and the center of the corresponding triangle "below" it. (c)

Connect each such "supplementary vertex" with its immediate neighbors above, below, and in the same plane. (d) Give all edge lengths. (e) Then the skeleton 3 -geometry between the two 2 -geometries is rigidly specified. It is made up of five types of tetrahedrons, as follows. (1) "Rightthrough blocks," such as

P P P^{'} S T

(six of these hinge on

P P^{'}

when

P

is a normal vertex; five, when it is a 5 -fold vertex, such as indicated by

a

at the upper right). (2) "Lower-facing blocks," such as

Q P T

. (3) "Lower-packing blocks," such as Q®ST .

(4, 5)

Corresponding "upper-facing blocks" and "upper-packing blocks" (not shown). The number of blocks of each kind is appropriately listed here for the two extreme cases of a 2-geometry that consists (a) of a normal hexagonal lattice extending indefinitely in a plane and (b) of a lattice consisting of the minimum number of 5 -fold vertices ("type

a

vertices") that will permit closeup into a 2 -sphere.

2-geometry of upper

(or lower) face

Hexagonal pattern

of triangles

Icosahedron

made of triangles

Its topology

Infinite 2-plane

2-sphere

Vertices on upper face

Nature of these vertices

6 -fold

5 -fold

Edge lengths on upper face

3 V

5 V = 30

Triangles on upper face

2 V

Number of "supplementary vertices"

2 V

Outer facing blocks

2 V

Outer packing blocks

3 V

Right through blocks

6 V

Inner packing blocks

3 V

Inner facing blocks

2 V

§42.6. THE CHOICE OF EDGE LENGTHS

So much for the lattice structure of the 4-geometry; now for the other issue, the freedom that exists in the choice of edge lengths. Why not make the simplest choice and let all edges be light rays? Because the 4 -geometry would not then be fully determined. The geometry

g_{α β} (x^{μ})

differs from the geometry

λ (x^{μ}) g_{α β} (x^{μ})

, even though the same points that are connected by light rays in the one geometry are also connected by light rays in the other geometry.
If none of the edges is null, it is nevertheless natural to take some of the edge lengths to be spacelike and some to be timelike. In consequence the area

A

of the triangle in some cases will be real, in other cases imaginary. In 3-space the parallelogram (double triangle) spanned by two vectors

B

and

C

is described by a vector

2 A = B \times C

perpendicular to the two vectors. One obtains the magnitude of

A

from the formula

4 A^{2} = B^{2} C^{2} - (B \cdot C)^{2}

In 4-space, let

B

and

C

be two edges of the triangle. Then, as in three dimensions,

2 A

is dual to the bivector built from

B

and

C

. In other words, if

B

goes in the

t

direction and

C

in the

z

direction, then

A

is a bivector lying in the

(x, y)

plane. Consequently its magnitude

A

is to be thought of as a real quantity. Therefore the appropriate formula for the area

A

is (Tullio Regge)

\begin{matrix} (42.10) & 4 A^{2} = (B \cdot C)^{2} - B^{2} C^{2} \end{matrix}

The quantity

A

is real when the deficit angle

δ

is real. Thus the geometrically important product

A δ

is also real.

The choice of edge lengths:
(1) choose some timelike, others spacelike

(2) choose timelike lengths comparable to spacelike lengths
(3) why some lengths must be chosen arbitrarily

Deficit angles in terms of edge lengths

Past applications of Regge calculus

When the hinge lies in the

(x, y)

plane, on the other hand, the quantity

A

is purely imaginary. In that instance a test vector taken around the cycle of simplexes that swing on this hinge has undergone change only in its

z

and

t

components; that is, it has experienced a Lorentz boost; that is, the deficit angle

δ

is also purely imaginary. So again the product

A δ

is a purely real quantity.
Turn now from character of edge lengths to magnitude of edge lengths. It is desirable that the elementary building blocks sample the curvatures of space in different directions on a roughly equal basis. In other words, it is desirable not to have long needle-shaped building blocks nor pancake-shaped tetrahedrons and simplexes. This natural requirement means that the step forward in time should be comparable to the steps "sidewise" in space. The very fact that one should have to state such a requirement brings out one circumstance that should have been obvious before: the "hinge equations"

\begin{matrix} (42.7) & \sum_{\begin{matrix} hinges thet \\ haveedgep \\ in common \end{matrix}} δ_{h} cotan θ_{p h} = 0 (p = 1, 2, \dots), \end{matrix}

though they are as numerous as the edges, cannot be regarded as adequate to determine all edge lengths. There are necessarily relations between these equations that keep them from being independent. The equations cannot determine all the details of the necessarily largely arbitrary skeletonization process. They cannot do so any more than the field equations of general relativity can determine the coordinate system. With a given pattern of vertices (four-dimensional generalization of drawings in Box 42.3), one still has (a) the option how close together one will take successive layers of the structure and (b) how one will distribute a given number of points in space on a given layer to achieve the maximum payoff in accuracy (greater density of points in regions of greater curvature). To prepare a practical computer program founded on Regge calculus, one has to supply the machine not only with the hinge equations and initial conditions, but also with definite algorithms to remove all the arbitrariness that resides in options (a) and (b).

Formulas from solid geometry and four-dimensional geometry, out of which to determine the necessary hyperdihedral angles

α

and the deficit angles

δ

in terms of edge lengths and nothing but edge lengths, are summarized by Wheeler (1964a, pp. 469,470 , and 490 ) and by C. Y. Wong (1971). Regge (1961) also gives a formula for the Riemann curvature tensor itself in terms of deficit angles and number of edges running in a given direction [see also Wheeler (1964a, p. 471)].

§42.7. PAST APPLICATIONS OF REGGE CALCULUS

Wong (1971) has applied Regge calculus to a problem where no time development shows itself, where the geometry can therefore be treated as static, and where in addition it is spherically symmetric. He determined the Schwarzschild and ReissnerNordstrøm geometries by the method of skeletonization. Consider successive spheres
surrounding the center of attraction. Wong approximates each as an icosahedron. The condition

^{(3)} R = 16 π (\binom{energy density}{on the 3-space})

(

$ 21.5

) gives a recursion relation that determines the dimension of each icosahedron in terms of the two preceding icosahedra. Errors in the skeleton representation of the exact geometry range from roughly 10 percent to less than 1 percent, depending on the method of analysis, the quantity under analysis, and the fineness of the subdivision.

Skeletonization of geometry is to be distinguished from mere rewriting of partial differential equations as difference equations. One has by now three illustrations that one can capitalize on skeletonization without fragmenting spacetime all the way to the level of individual simplexes. The first illustration is the first part of Wong's work, where the time dimension never explicitly makes an appearance, so that the building blocks are three-dimensional only. The second is an alternative treatment, also given by Wong, that goes beyond the symmetry in

t

to take account of the symmetry in

θ

and

ϕ

. It divides space into spherical shells, in each of which the geometry is "pseudo-flat" in much the same sense that the geometry of a paper cone is flat. The third is the numerical solution for the gravitational collapse of a spherical star by May and White (1966), in which there is symmetry in

θ

and

ϕ

, but not in

r

t

. This zoning takes place exclusively in the

r, t

-plane. Each zone is a spherical shell. The difference as compared to Regge calculus (flat geometry within each building block) is the adjustable "conicity" given to each shell. The examples show that the decision about skeletonizing the geometry in a calculation is ordinarily not "whether" but "how much."

§42.8. THE FUTURE OF REGGE CALCULUS

In summary, Regge's skeleton calculus puts within the reach of computation problems that in practical terms are beyond the power of normal analytical methods. It affords any desired level of accuracy by sufficiently fine subdivision of the spacetime region under consideration. By way of its numbered building blocks, it also offers a practical way to display the results of such calculations. Finally, one can hope that Regge's truly geometric way of formulating general relativity will someday make the content of the Einstein field equations (Cartan's "moment of rotation"; see Chapter 15) stand out sharp and clear, and unveil the geometric significance of the so-called "geometrodynamic field momentum" (analysis of the boundary-value problem associated with the variational problem of general relativity in Regge calculus; see §21.12).

cmuma 43

SUPERSPACE: ARENA FOR THE DYNAMICS OF GEOMETRY

Traveler, there are no paths.
Paths are made by walking.
ANTONIO MACHADO (1940)

This chapter is entirely Track 2. Chapter 21 (initial-value formalism) is needed as preparation for it. In reading it, one will be helped by Chapter 42 (Regge calculus). It is not needed as preparation for any later chapter, but it will be helpful in Chapter 44 (vision of the future).

Superspace is the arena for geometrodynamics

§43.1. SPACE, SUPERSPACE, AND SPACETIME DISTINGUISHED

Superspace [Wheeler (1964a), pp. 459 ff ] is the arena of geometrodynamics. The dynamics of Einstein's curved space geometry runs its course in superspace as the dynamics of a particle unfolds in spacetime. This chapter gives one simple version of superspace, and a little impression of alternative versions of superspace that also have their uses. It describes the classical dynamics of geometry in superspace in terms of the Hamilton-Jacobi principle of Boxes 25.3 and 25.4. No version of mechanics makes any shorter the leap from classical dynamics to quantum. Thus it provides a principle ("Einstein-Hamilton-Jacobi or EHJ equation") for the propagation of wave crests in superspace, and for finding where those wave crests give one the classical equivalent of constructive interference ("envelope formation"). In this way one finds the track of development of 3-geometry with time expressed as a sharp, thin "leaf of history" that slices through superspace. The quantum principle replaces this deterministic account with a fuzzed-out leaf of history of finite thickness. In consequence, quantum fluctuations take place in the geometry of space that dominate the scene at distances of the order of the Planck length,

L^{*} = {(ℏ G / c^{3})}^{1 / 2} = 1.6 \times

10^{- 33} cm

, and less. The present survey simplifies by considering only the dynamics of curved empty space. When sources are present and are to be taken into account, supplementary terms are to be added, some of the literature on which is listed.

In all the difficult investigations that led in the course of half a century to some understanding of the dynamics of geometry, both classical and quantum, the most

Box 43.1 GEOMETRODYNAMICS COMPARED WITH PARTICLE DYNAMICS

Concept	Particle dynamics	Geometrodynamics
Dynamic entity	Particle	Space
Descriptors of momentary configuration	$x, t$ ("event")	(3) $%$ ("3-geometry")
Classical history	$x = x (t)$	(4) $%$ ("4-geometry")
History is a stockpile of configurations?	Yes. Every point on world line gives a momentary configuration of particle	Yes. Every spacelike slice through $^{(4)}$ ) gives a momentary configuration of space
Dynamic arena	Spacetime (totality of all points $x, t$ )	Superspace (totality of all (3) E's' $^{'}$

difficult point was also the simplest: The dynamic object is not spacetime. It is space. The geometric configuration of space changes with time. But it is space, three-dimensional space, that does the changing (see Box 43.1).
Space will be treated here as "closed" or, in mathematical language, "compact," either because physics adds to Einstein's second-order differential equations the requirement of closure as a necessary and appropriate boundary condition [Einstein (1934, p. 52; 1950); Wheeler (1959; 1964c). Hönl (1962); see also §21.12] or because that requirement simplifies the mathematical analysis, or for both reasons together.

One can approximate a smooth, closed 3-geometry by a skeleton 3-geometry built out of tetrahedrons, as indicated schematically in Figure 43.1 (see Chapter 42 on the Regge calculus). Specify the 98 edge-lengths in this example to fix all the features of the geometry; and fix these 98 edge-lengths by giving the location of a single point in a space of 98 dimensions. This 98 -dimensional manifold, this "truncated superspace," goes over into superspace [Wheeler (1964a), pp. 453, 459, 463, 495] in the idealization in which the tracer points increase in density of coverage without limit. Accounts of superspace with more mathematical detail are given by DeWitt (1967a,b), Wheeler (1970), and Fischer (1970).

Let the representative point move from one location to a nearby location, either in truncated superspace or in full superspace. Then all edge-lengths alter, and the 3-geometry of Figure 43.1 moves as if alive. No better illustration can one easily supply of what it means to speak of the "dynamics of space."

The term "3-geometry" makes sense as well in quantum geometrodynamics as in classical theory. So does superspace. But spacetime does not. Give a 3-geometry, and give its time rate of change. That is enough, under typical circumstances (see Chapter 21) to fix the whole time-evolution of the geometry; enough in other words, to determine the entire four-dimensional spacetime geometry, provided one is

3 -geometry is the dynamic object

Finite-dimensional "truncated superspace"

Figure 43.1.
Superspace in the simplicial approximation. Upper left, space (depicted as two-dimensional but actually three-dimensional). Upper right, simplical approximation to space. The approximation can be made arbitrarily good by going to the limit of an arbitrarily fine decomposition. The curvature at a typical location is measured by a deficit angle. This angle is completely determined by the edge lengths (

L_{1}

L_{2}, \dots L_{8}

in the figure) of the simplexes that meet at that location. When there are 98 edge lengths altogether in the simplicial representation of the geometry, then this geometry is completely specified by a single point in a 98 -dimensional space (lower diagram; "superspace").

The concept of spacetime is incompatible with the quantum principle
considering the problem in the context of classical physics. In the real world of quantum physics, however, one cannot give both a dynamic variable and its time-rate of change. The principle of complementarity forbids. Given the precise 3-geometry at one instant, one cannot also know at that instant the time-rate of change of the 3 -geometry. In other words, given the geometrodynamic field coordinate, one cannot know the geometrodynamic field momentum. If one assigns the intrinsic 3-geometry, one cannot also specify the extrinsic curvature.
The uncertainty principle thus deprives one of any way whatsoever to predict, or even to give meaning to, "the deterministic classical history of space evolving
in time." No prediction of spacetime, therefore no meaning for spacetime, is the verdict of the quantum principle. That object which is central to all of classical general relativity, the four-dimensional spacetime geometry, simply does not exist, except in a classical approximation.

These considerations reveal that the concepts of spacetime and time are not primary but secondary ideas in the structure of physical theory. These concepts are valid in the classical approximation. However, they have neither meaning nor application under circumstances where quantum geometrodynamic effects become important. Then one has to forego that view of nature in which every event, past, present, or future, occupies its preordained position in a grand catalog called "spacetime," with the Einstein interval from each event to its neighbor eternally established. There is no spacetime, there is no time, there is no before, there is no after. The question of what happens "next" is without meaning.

That spacetime is not the right way does not mean there is no right way to describe the dynamics of geometry consistent with the quantum principle. Superspace is the key to one right way to describe the dynamics (see Figure 43.2).

Figure 43.2.
Space, spacetime, and superspace. Upper left: Five sample configurations,

A, B, C, D, E

, attained by space in the course of its expansion and recontraction. Below: Superspace and these five sample configurations, each represented by a point in superspace. Upper right: Spacetime. A spacelike cut, like

A

, through spacetime gives a momentary configuration of space. There is no compulsion to limit attention to a one-parameter family of slices,

A, B, C, D, E

through spacetime. The phrase "many-fingered time" is a slogan telling one not to so limit one's slices, and

B^{'}

is an example of this freedom in action. The 3-geometries

B^{'}

and

A, B, C, D, E

, like all 3-geometries obtained by all spacelike slices whatsoever through the given classical spacetime, lie on a single bent leaf of history, indicated in the diagram, and cutting its thin slice through superspace. A different spacetime, in other words, a different solution of Einstein's field equation, means a different leaf of history (not indicated) slicing through superspace.

§43.2. THE DYNAMICS OF GEOMETRY DESCRIBED IN THE LANGUAGE OF THE SUPERSPACE OF THE $^{(3)} y^{'}$ 's

Spacetime is a classical leaf of history slicing through superspace

Given a spacetime, one can construct the corresponding leaf of history slicing through superspace. Conversely, given the leaf of history, one can reconstruct the spacetime.

Consider the child's toy commonly known as "Chinese boxes." One opens the outermost box only to reveal another box; when the second box is opened, there is another box, and so on, until eventually there are dozens of boxes scattered over the floor. Or conversely the boxes can be put back together, nested one inside the other, to reconstitute the original package. The packaging of

^{(3)} y^{'}

s into a

^{(4)} y

is much more sophisticated. Nature provides no monotonic ordering of the

^{(3)} y^{'}

's. Two of the dynamically allowed

^{(3)} ξ

's, taken at random, will often cross each other one or more times. When one shakes the

^{(4)} y

apart, one therefore gets enormously more

^{(3)} y^{'}

's "spread out over the floor" than might have been imagined. Conversely, when one puts back together all of the

^{(3)}

S's lying on the leaf of history, one gets a structure with a rigidity that might not otherwise have been foreseen. This rigidity arises from the infinitely rich interleaving and intercrossing of clear-cut, well-defined

^{(3)}

g's one with another.

In summary: (1) Classical geometrodynamics in principle constitutes a device, an algorithm, a rule for calculating and constructing a leaf of history that slices through superspace. (2) The

^{(3)}

'g's that lie on this leaf of history are YES 3-geometries; the vastly more numerous

^{(3)} y

's that do not are NO 3-geometries. (3) The YES

^{(3)} y

's are the building blocks of the

^{(4)} y

that is classical spacetime. (4) The interweaving and interconnections of these building blocks give the

^{(4)} y

its existence, its dimensionality, and its structure. (5) In this structure every

^{(3)} y

has a rigidly fixed location of its own. (6) In this sense one can say that the "many-fingered time" of each 3-geometry is specified by the very interlocking construction itself. Baierlein, Sharp and Wheeler (1962) say a little more on this concept of "3-geometry as carrier of information about time."

How different from the textbook concept of spacetime! There the geometry of spacetime is conceived as constructed out of elementary objects, or points, known as "events." Here, by contrast, the primary concept is 3-geometry, in abstracto, and out of it is derived the idea of event. Thus, (1) the event lies at the intersection of such and such

^{(3)} Θ^{'}

's; and (2) it has a timelike relation to (earlier or later than, or synchronous with) some other

^{(3)} ℓ

, which in turn (3) derives from the intercrossings of all the other

^{(3)}

g's.

When one turns from classical theory to quantum theory, one gives up the concept of spacetime, except in the semiclassical approximation. Therefore, one gives up any immediate possibility whatsoever of defining the concept, normally regarded as so elemental, of an "event." The theory itself, however, here as always [Bohr and Rosenfeld (1933)], defines in and by itself, in its own natural way, the procedures-in-principle for measuring all those quantities that are in principle measurable.

Quantum theory upsets the sharp distinction between YES 3-geometries and NO

3-geometries. It assigns to each 3-geometry not a YES or a NO, but a probability amplitude,

\begin{matrix} (43.1) & ψ = ψ (^{(3)} ξ) \end{matrix}

This probability amplitude is highest near the classically forecast leaf of history and falls off steeply outside a zone of finite thickness extending a little way on either side of the leaf.

Were one to take, instead of a physically relevant probability amplitude function, a typical solution of the relevant wave equation, one would have to expect to see not one trace of anything like classical geometrodynamics. The typical probability amplitude function is spread all over superspace. No surprise! Already in classical theory one has to reckon with a Hamilton-Jacobi function,

\begin{matrix} (43.2) & S = S (^{(3)} ξ) \end{matrix}

spread out over superspace. Moreover, this "dynamic phase function" of classical geometrodynamics gives at once the phase of

ψ

, according to the formula

\begin{matrix} (43.3) & ψ (^{(3)} y) = (\binom{slowly varying}{amplitude function}) e^{(i / ℏ) S (^{(3)} ℓ)} \end{matrix}

indication enough that

ψ

and

S

are both unlocalized.
Dynamics first clearly becomes recognizable when sufficiently many such spreadout probability amplitude functions are superposed to build up a localized wave packet, as in the elementary examples of Boxes 25.3 and 25.4 ; thus,

\begin{matrix} (43.4) & ψ = c_{1} ψ_{1} + c_{2} ψ_{2} + \dots \end{matrix}

Constructive interference occurs where the phases of the several individual waves agree:

\begin{matrix} (43.5) & S_{1} (^{(3)} ξ) = S_{2} (^{(3)} ξ) = \dots \end{matrix}

This is the condition that distinguishes YES 3-geometries from NO 3-geometries. It is the tool for constructing a leaf of history in superspace. It is the key to the dynamics of geometry. Moreover, it is an equation that says not one word about the quantum principle. It is not surprising that the equation of constructive interference in (43.5) makes the leap from classical theory to quantum theory the shortest.

§43.3. THE EINSTEIN-HAMILTON-JACOBI EQUATION

Should one write down a differential equation for the Hamilton-Jacobi function

S (^{(3)} ϑ)

, solve it, and then analyze the properties of the solution? The exact opposite is simpler: look at the properties of the solution, and from that inspection find out what equation the dynamic phase or action

S

must satisfy.

Probability amplitude for a 3-geometry

Wave packet recovers classical geometrodynamics

Hilberts' principle of least action reads

\begin{matrix} (43.6) & I_{Hilbert} = (1 / 16 π) \int^{(4)} R (- g)^{1 / 2} d^{4} x = extremum. \end{matrix}

After one separates off complete derivatives in the integrand, what is left [see equations (21.13) and (21.95)] becomes

\begin{aligned} (1 / 16 π) I_{ADM} = & I_{true} = (1 / 16 π) \int {π^{i j} \partial g_{i j} / \partial t + N g^{1 / 2} R \\ (43.7) & + N g^{- 1 / 2} [\frac{1}{2} (Tr π)^{2} - Tr (Π^{2})] + 2 N_{i} π^{i j}_{∣ j}} d^{4} x \end{aligned}

In (43.7), but not in (43.6),

g

stands for the determinant of the three-dimensional metric tensor,

g_{i j}

, and

R

for the scalar curvature invariant of the 3-geometry; the suffix

^{(3)}

is omitted for simplicity. The integral is extended from (1) a spacelike hypersurface on which a 3-geometry is given with metric

g_{i j}^{'} (x, y, z)

to (2) a spacelike hypersurface on which a 3 -geometry is given with metric

g_{i j}^{''} (x, y, z)

. Whatever is adjustable in the chunk of spacetime between is now to be considered as having been adjusted to extremize the integral. Therefore the value of the integral

I_{ADM}

becomes a functional of the metrics on the two hypersurfaces and nothing more.

Next, holding fixed the metric

g_{i j}^{'} (x, y, z)

on the earlier hypersurface, change slightly or even more than slightly the metric on the later hypersurface. Solve the new variation problem and get a new value of

I_{ADM}

. Proceeding further in this way, for each new

g_{i j}^{''}

one gets a new value of

I_{ADM}

. Call the functional

I_{ADM}

of the metric defined in this way "Hamilton's principal function," or the "action" or the "dynamic path length,*"

S (g_{i j} (x, y, z))

of the "history of geometry" that connects the two given 3-geometries. The double prime suffix is dropped from

g_{i j}^{''}

here and hereafter to simplify the notation. One knows from other branches of mechanics that the quantity defined in this way,

S (g_{i j})

, when it exists, even though it is a special solution, nevertheless is always a solution of the Hamilton-Jacobi equation. Jacobi could look for more general solutions, but Hamilton already had one!

For (43.7) to be an extremal with respect to variations of the lapse

N

and the shift components

N_{i}

, it was necessary (see Chapter 21) that the coefficients of these four quantities should vanish; thus,

\begin{matrix} (43.8) & g^{- 1 / 2} [\frac{1}{2} (Tr π)^{2} - Tr π^{2}] + g^{1 / 2} R = 0 \end{matrix}

and

\begin{matrix} (43.9) & π^{i j}_{1 j} = 0 . \end{matrix}

In the expression for the extremal value of the action, only one term, the first, is left:

\begin{matrix} (43.10) & S (g (x, y, z)) = I_{ADM, extremal} = \int_{g_{i j}^{'}}^{g_{i j}} {π^{i j} \partial g_{i j} / \partial t} d^{4} x \end{matrix}

The effect of a slight change,

δ g_{i j}

, in the 3-metric at the upper limit is therefore easy to read off:

\begin{matrix} (43.11) & δ S = \int π^{i j} (x, y, z) δ g_{i j} (x, y, z) d^{3} x \end{matrix}

The language of "functional derivative" [see, for example, Bogoliubov and Shirkov (1959)] allows one to speak in terms of a derivative rather than an integral:

\begin{matrix} (43.12) & \frac{δ S}{δ g_{i j}} = π^{i j} \end{matrix}

The "field momenta" acquire a simple meaning: they give the rate of change of the action with respect to the continuous infinitude of "field coordinates,"

g_{i j} (x, y, z)

. (Here the

x, y, z

, as well as the

i

and

j

, serve as mere labels.)

Although the phase function

S

appears to depend on all six metric coefficients

g_{i j}

individually, it depends in actuality only on that combination of the

g_{i j}

which is locked to the 3 -geometry. To verify this point, express a particular 3-geometry

^{(3)}

) throughout one local coordinate patch in terms of one set of coordinates

x^{p}

by one set of metric coefficients

g_{p q}

. Reexpress the same 3 -geometry in terms of coordinates

{\bar{x}}^{p}

shifted by the small amount

ξ^{p}

\begin{matrix} (43.13) & {\bar{x}}^{p} = x^{p} - ξ^{p} . \end{matrix}

To keep the 3-geometry the same, that is, to keep unchanged the distance

d s

from one coordinate-independent point to another, the metric coefficients have to change:

\begin{matrix} (43.14) & {\bar{g}}_{p q} = g_{p q} + ξ_{p ∣ q} + ξ_{q ∣ p} \end{matrix}

Let the phase function

S

(or in quantum mechanics, let the probability amplitude

ψ)

be considered to be expressed as a functional of the metric coefficients

g_{11} (x)

g_{12} (x), \dots, g_{33} (x)

. Changes

δ g_{p q} (x)

in these coefficients alter the H-J phase function and the probability amplitude by the amounts

\begin{aligned} δ S & = \int (δ S / δ g_{p q}) δ g_{p q} d^{3} x \\ (43.15) & δ ψ & = \int (δ ψ / δ g_{p q}) δ g_{p q} d^{3} x \end{aligned}

according to the standard definition of functional derivative. Therefore the coordinate change produces an ostensible change in the dynamic path length or phase

S

given by

\begin{aligned} (43.16) & δ S & = \int (δ S / δ g_{p q}) (ξ_{p ∣ q} + ξ_{q ∣ p}) d^{3} x \\ = - 2 \int {(δ S / δ g_{p q})}_{∣ q} ξ_{p} d^{3} x . \end{aligned}

This change must vanish if

S

is to depend on the 3-geometry alone, and not on
the coordinates in terms of which that 3-geometry is expressed; and must vanish, moreover, for arbitrary choice of the

ξ_{p}

. From this condition, one concludes

\begin{matrix} (43.17) & {(\frac{δ S}{δ g_{p q}})}_{∣ q} = 0 . \end{matrix}

Likewise, one finds three equations on the wave function

ψ

itself, as distinguished from its phase

S / ℏ

; thus,

\begin{matrix} (43.18) & {(\frac{δ ψ}{δ g_{p q}})}_{∣ q} = 0 . \end{matrix}

But (43.17), by virtue of (43.12), is identical with (43.9). In this sense (43.9) merely verifies what one already knew had to be true: the classical Hamilton-Jacobi function

S

(like the probability amplitude function

ψ

of quantum theory) depends on 3-geometry, not on individual metric coefficients, and not on choice of coordinates.
All the dynamic content of geometrodynamics is summarized in the sole remaining equation (43.8), which takes the form

\begin{matrix} (43.19) & g^{- 1 / 2} [\frac{1}{2} g_{p q} g_{r s} - g_{p r} g_{q s}] \frac{δ S}{δ g_{p q}} \frac{δ S}{δ g_{r s}} + g^{1 / 2} R = 0 \end{matrix}

This is the Einstein-Hamilton-Jacobi equation, first given explicitly in the literature by Peres (1962) on the foundation of earlier work by himself and others on the Hamiltonian formulation of geometrodynamics. This equation tells how fronts of constant

S

("wave crests") propagate in superspace.

That the one EHJ equation (43.19) contains as much information as all ten components of Einstein's field equation has been demonstrated by Gerlach (1969). The central point in the analysis is the principle of constructive interference, and the main requirement for a proper treatment of this point is the concept of a completely parametrized solution of the EHJ equation.

The problem of a particle moving in two-dimensional space, as treated by the Hamilton-Jacobi method in Boxes 25.3 and 25.4, required for complete analysis a solution containing two distinct and independently adjustable parameters, the energy per unit mass,

\tilde{E}

, and angular momentum per unit mass,

\tilde{L}

; thus

\begin{aligned} S (r, θ, t; \tilde{E}, \tilde{L}) = - \tilde{E} t + \tilde{L} θ \\ (43.20) & + \int^{r} {[{\tilde{E}}^{2} - (1 - 2 M / r) (1 + {\tilde{L}}^{2} / r^{2})]}^{1 / 2} \frac{d r}{(1 - 2 M / r)} + δ (\tilde{E}, \tilde{L}) \end{aligned}

Here the additive phase

δ (\tilde{E}, \tilde{L})

is required if one is to be able to arrange for the particle to arrive at a given

r

-value at a specified

t

value and at a specified value of

θ

. One thinks of superposing four probability amplitudes, as in (43.4), with dynamic phases,

S

, given by

(43.20)

and the parameters taking on, respectively, the following four sets of values:

(\tilde{E}, \tilde{L}); (\tilde{E} + Δ \tilde{E}, \tilde{L}); (\tilde{E}, \tilde{L} + Δ \tilde{L})

; and

(\tilde{E} + Δ \tilde{E}

\tilde{L} + Δ \tilde{L})

. The principle of constructive interference leads to the conditions

\begin{aligned} \partial S / \partial \tilde{E} = 0 \\ (43.21) & \partial S / \partial \tilde{L} = 0 . \end{aligned}

The points in the spacetime

(r, θ, t)

that satisfy these conditions are the YES points; they lie on the world line. The ones that don't are the NO points.

The desired solution of the EHJ equation (43.19) contains not two parameters (plus an additive phase,

δ

, depending on these two parameters), but an infinity of parameters, and even a continuous infinity of parameters. Thus the parameters are not to be designated as

α_{1}, α_{2}, \dots; β_{1}, β_{2}, \dots

(parameters labeled by a discrete index), but as

α (u, v, w)

and

β (u, v, w)

(two parameters "labeled" by three continuous indices

u, v, w

). Accidentally omit one of this infinitude of parameters? How could one ever hope to know that what purported to be a complete solution of the EHJ equation was not in actuality complete? Happily Gerlach provides a procedure to test the parameters for completeness.

Granted completeness, Gerlach goes on to show that the "leaf of history in superspace" or collection of 3-geometries that satisfy the conditions of constructive interference,

\begin{aligned} \frac{δ S (^{(3)} ξ; α (u, v, w), β (u, v, w))}{δ α} = 0, \\ (43.22) & \frac{δ S (^{(3)};; α (u, v, w), β (u, v, w))}{δ β} = 0, \end{aligned}

is identical with the leaf of history, or equivalent 4 -geometry, given by the ten components of Einstein's geometrodynamic law.

From the Hamilton-Jacobi equation for a problem in elementary mechanics, it is a short step to the corresponding Schroedinger equation; similarly in geometrodynamics. No one has done more than Bryce DeWitt to explore the meaning and consequences of this "Einstein-Schroedinger equation" [DeWitt (1967a,b)]. One of the most interesting consequences is the existence of a conserved current in superspace, analogous to the conserved current

j_{μ} = \frac{ℏ}{2 i m} (ψ^{*} \frac{\partial ψ}{\partial x^{μ}} - ψ \frac{\partial ψ^{*}}{\partial x^{μ}})

that one encounters in the Klein-Gordon wave equation for a particle of spin zero.
It is an unhappy feature of this Einstein-Schroedinger wave equation that it contains second derivatives; so one has to specify both the probability amplitude, and the normal derivative of the probability amplitude, on the appropriate "super-

Condition of constructive interference gives classical "leaf of history" or spacetime
hypersurface" in superspace, in order to be able to predict the evolution of this state function elsewhere in superspace. One suggested way out of this situation-it is at least an inconvenience, possibly a real difficulty-has been proposed by Leutwyler (1968): impose a natural boundary condition that reduces the number of independent solutions from the number characteristic of a second-order equation to the number characteristic of a first-order equation. Another way out is to formulate the dynamics quite differently, in the way proposed by Kuchař (see Chapter 21), in which the resulting equation is of first order in the variable analogous to time.
The exploration of quantum geometrodynamics is simplified when one treats most of the degrees of freedom of the geometry as frozen out, by imposition of a high degree of symmetry. Then one is left with one, two, or three degrees of freedom (see Chapter 30, on mixmaster cosmology), or even infinitely many, and is led to manageable problems of quantum mechanics [Misner (1972a, 1973)].

§43.4. FLUCTUATIONS IN GEOMETRY

Of all the remarkable developments of physics since World War II, none is more impressive than the prediction and verification of the effects of the vacuum fluctuations in the electromagnetic field on the motion of the electron in the hydrogen atom (Figure 43.3). That development made it impossible to overlook the effects of such fluctuations throughout all physics and, not least, in the geometry of spacetime itself.

Figure 43.3.
Symbolic representation of motion of electron in hydrogen atom as affected by fluctuations in electric field in vacuum ("vacuum" or "ground state" or "zero-point" fluctuations). The electric field associated with the fluctuation,

E_{x} (t) = \int E_{x} (ω) e^{- i ω t} d ω

, adds to the static electric field provided by the nucleus itself. The additional field brings about in the most elementary approximation the displacement

Δ x = \int (e / m ω^{2}) E_{x} (ω) e^{- i ω t} d ω

. The average vanishes but the root mean square

⟨ (Δ x)^{2} ⟩

does not. In consequence the electron feels an effective atomic potential altered from the expected value

V (x, y, z)

by the amount

Δ V (x, y, z) = \frac{1}{2} ⟨ (Δ x)^{2} ⟩ \nabla^{2} V (x, y, z)

The average of this perturbation over the unperturbed motion accounts for the major part of the observed Lamb-Retherford shift

Δ E = ⟨ Δ V (x, y, z) ⟩

in the energy level. Conversely, the observation of the expected shift makes the reality of the vacuum fluctuations inescapably evident.

From the zero-point fluctuations of a single oscillator to the fluctuations of the electromagnetic field to geometrodynamic fluctuations is a natural order of progression.

A harmonic oscillator in its ground state has a probability amplitude of

\begin{matrix} (43.23) & ψ (x) = {(\frac{m ω}{π ℏ})}^{1 / 4} e^{- (m ω / 2 h) x^{2}} \end{matrix}

to be displaced by the distance

x

from its natural classical position of equilibrium. In this sense, it may be said to "resonate" or "fluctuate" between locations in space ranging over a region of extent

\begin{matrix} (43.24) & Δ x \sim (ℏ / m ω)^{1 / 2} . \end{matrix}

The electromagnetic field can be treated as an infinite collection of independent "field oscillators," with amplitudes

ξ_{1}, ξ_{2}, \dots

. When the Maxwell field is in its state of lowest energy, the probability amplitude-for the first oscillator to have amplitude

ξ_{1}

, and simultaneously the second oscillator to have amplitude

ξ_{2}

, the third

ξ_{3}

, and so on-is the product of functions of the form (43.23), one for each oscillator. When the scale of amplitudes for each oscillator is suitably normalized, the resulting infinite product takes the form

\begin{matrix} (43.25) & ψ (ξ_{1}, ξ_{2}, \dots) = N \exp [- (ξ_{1}^{2} + ξ_{2}^{2} + \dots)] \end{matrix}

This expression gives the probability amplitude

ψ

for a configuration

B (x, y, z)

of the magnetic field that is described by the Fourier coefficients

ξ_{1}, ξ_{2}, \dots

. One can forgo any mention of these Fourier coefficients if one desires, however, and rewrite (43.25) directly in terms of the magnetic field configuration itself [Wheeler (1962)]:

\begin{matrix} (43.26) & ψ (B (x, y, z)) = r \exp (- \iint \frac{B (x_{1}) \cdot B (x_{2})}{16 π^{3} ℏ c r_{12}^{2}} d^{3} x_{1} d^{3} x_{2}) . \end{matrix}

One no longer speaks of "the" magnetic field, but instead of the probability of this, that, or the other configuration of the magnetic field, even under circumstances, as here, where the electromagnetic field is in its ground state. [See Kuchař (1970) for a similar expression for the "ground state" functional of the linearized gravitational field.]

It is reasonable enough under these circumstances that the configuration of greatest probability is

B (x, y, z) = 0

. Consider for comparison a configuration where the magnetic field is again everywhere zero except in a region of dimension

L

. There let the field, subject as always to the condition

div B = 0

, be of the order of magnitude

Δ B

. The probability amplitude for this configuration will be reduced relative to the nil configuration by a factor

\exp (- I)

. Here the quantity

I

in the exponent is of the order

(Δ B)^{2} L^{4} / ℏ c

. Configurations for which

I

is large compared to 1 occur with negligible probability. Configurations for which

I

is small compared to 1 occur with practically the same probability as the nil configuration. In this sense, one can

Fluctuations for oscillator and for electromagnetic field
say that the fluctuations in the magnetic field in a region of extension

L

are of the order of magnitude

\begin{matrix} (43.27) & Δ B \sim \frac{(ℏ c)^{1 / 2}}{L^{2}} \end{matrix}

In other words, the field "resonates" between one configuration and another with the range of configurations of significance given by (43.27). Moreover, the smaller is the region of space under consideration, the larger are the field magnitudes that occur with appreciable probability.

Still another familiar way of speaking about electromagnetic field fluctuations gives additional insight relevant to geometrodynamics. One considers a measuring device responsive in comparable measure to the magnetic field at all points in a region of dimension

L

. One asks for the effect on this device of electromagnetic disturbances of various wavelengths. A disturbance of wavelength short compared to

L

will cause forces to act one way in some parts of the detector, and will give rise to nearly compensating forces in other parts of it. In contrast, a disturbance of a long wavelength

λ

produces forces everywhere in the same direction, but of a magnitude too low to have much effect. Thus the field, estimated from the equation

(\begin{array}{l} energy of electromagnetic \\ wave of wavelength λ in a \\ domain of volume λ^{3} \end{array}) \sim (\binom{energy of one quantum}{of wavelength λ})

B^{2} λ^{3} \sim \frac{ℏ c}{λ}

\begin{matrix} (43.28) & B \sim \frac{(ℏ c)^{1 / 2}}{λ^{2}} \end{matrix}

is very small if

λ

is large compared to the domain size

L

. The biggest effect is caused by a disturbance of wavelength

λ

comparable to

L

itself. This line of reasoning leads directly from (43.28) to the standard fluctuation formula (43.27).

Similar considerations apply in geometrodynamics. Quantum fluctuations in the geometry are superposed on and coexist with the large-scale, slowly varying curvature predicted by classical deterministic general relativity. Thus, in a region of dimension

L

, where in a local Lorentz frame the normal values of the metric coefficients will be

- 1, 1, 1, 1

, there will occur fluctuations in these coefficients of the order

\begin{matrix} (43.29) & Δ g \sim \frac{L^{*}}{L} \end{matrix}

fluctuations in the first derivatives of the

g_{i k}

's of the order

\begin{matrix} (43.30) & Δ Γ \sim \frac{Δ g}{L} \sim \frac{L^{*}}{L^{2}} \end{matrix}

and fluctuations in the curvature of space of the order

\begin{matrix} (43.31) & Δ R \sim \frac{Δ g}{L^{2}} \sim \frac{L^{*}}{L^{3}} \end{matrix}

Here

\begin{matrix} (43.32) & L^{*} = {(\frac{ℏ G}{c^{3}})}^{1 / 2} = 1.6 \times 10^{- 33} cm \end{matrix}

is the so-called Planck length [Planck (1899)].
It is appropriate to look at orders of magnitude. The curvature of space within and near the earth, according to classical Einstein theory, is of the order

\begin{aligned} R & \sim (\frac{G}{c^{2}}) ρ \sim (0.7 \times 10^{- 28} cm / g) (5 g / {cm}^{3}) \\ (43.33) & \sim 4 \times 10^{- 28} {cm}^{- 2} \end{aligned}

This quantity has a very direct physical significance. It measures the "tide-producing component of the gravitational field" as sensed, for example, in a freely falling elevator or in a spaceship in free orbit around the earth. By comparison, the quantum fluctuations in the curvature of space are only

\begin{matrix} (43.34) & Δ R \sim 10^{- 33} {cm}^{- 2} \end{matrix}

even in a domain of observation as small as 1 cm in extent. Thus the quantum fluctuations in the geometry of space are completely negligible under everyday circumstances.

Even in atomic and nuclear physics the fluctuations in the metric,

Δ g \sim \frac{10^{- 33} cm}{10^{- 8} cm} \sim 10^{- 25}

and

\begin{matrix} (43.35) & Δ g \sim \frac{10^{- 33} cm}{10^{- 13} cm} \sim 10^{- 20} \end{matrix}

are so small that it is completely in order to idealize the physics as taking place in a flat Lorentzian spacetime manifold.

The quantum fluctuations in the geometry are nevertheless inescapable, if one is to believe the quantum principle and Einstein's theory. They coexist with the geometrodynamic development predicted by classical general relativity. The fluctuations widen the narrow swathe cut through superspace by the classical history of the geometry. In other words, the geometry is not deterministic, even though it looks so at the everyday scale of observation. Instead, at a submicroscopic scale it "resonates" between one configuration and another and another. This terminology means no more and no less than the following: (1) Each configuration

^{(3)}

& has its own probability amplitude

ψ = ψ (^{(3)} ℓ)

. (2) These probability amplitudes have comparable magnitudes for a whole range of 3-geometries included within the limits (43.29) on
either side of the classical swathe through superspace. (3) This range of 3-geometries is far too variegated on the submicroscopic sale to fit into any one 4 -geometry, or any one classical geometrodynamic history. (4) Only when one overlooks these small-scale fluctuations (

\sim 10^{- 33} cm

) and examines the larger-scale features of the 3-geometries do they appear to fit into a single space-time manifold, such as comports with the classical field equations.

These small-scale fluctuations tell one that something like gravitational collapse is taking place everywhere in space and all the time; that gravitational collapse is in effect perpetually being done and undone; that in addition to the gravitational collapse of the universe, and of a star, one has also to deal with a third and, because it is constantly being undone, most significant level of gravitational collapse at the Planck scale of distances.

EXERCISES

Exercise 43.1. THE ACTION PRINCIPLE FOR A FREE PARTICLE IN NONRELATIVISTIC MECHANICS

Taking as action principle

I = \int L d t =

extremum, with specified

x^{'}, t^{'}

and

x^{''}, t^{''}

at the two limits, and with

L = \frac{1}{2} m (d x / d t)^{2}

, find (1) the extremizing history

x = x (t)

and (2) the dynamical path length or action

S (x^{''}, t^{''}; x^{'}, t^{'}) = I_{extremum}

in its dependence on the end points. Also (3) write down the Hamilton-Jacobi equation for this problem, and (4) verify that

S (x, t

;

x^{'}, t^{'})

satisfies this equation. Finally, imagining the Hamilton-Jacobi equation not to be known, (5) derive it from the already known properties of the function

S

itself.

Exercise 43.2. THE ACTION FOR THE HARMONIC OSCILLATOR

The kinetic energy is

\frac{1}{2} m (d x / d t)^{2}

and the potential energy is

\frac{1}{2} m ω^{2} x^{2}

. Carry through the analysis of parts (1), (2), (3), (4) of the preceding exercise. Partial answer:

S = \frac{m ω}{2} \frac{(x^{2} + x^{' 2}) \cos ω (t - t^{'}) - 2 x x^{'}}{\sin ω (t - t^{'})}

Verify that

\partial S / \partial x

gives momentum and

- \partial S / \partial t

gives energy.

Exercise 43.3. QUANTUM PROPAGATOR FOR HARMONIC OSCILLATOR

Show that the probability amplitude for a simple harmonic oscillator to transit from

x^{'}, t^{'}

x^{''}, t^{''}

\begin{aligned} ⟨ x^{''}, t^{''}; x^{'}, t^{'} ⟩ \\ = {(\frac{m ω}{2 π i ℏ \sin ω (t^{''} - t^{'})})}^{1 / 2} \times \exp \frac{i m ω [(x^{'' 2} + x^{' 2}) \cos ω (t^{''} - t^{'}) - 2 x^{''} x^{'}]}{2 ℏ \sin ω (t^{''} - t^{'})}, \end{aligned}

and that it reduces for the case of a free particle to

⟨ x^{''}, t^{''}; x^{'}, t^{'} ⟩ = {(\frac{m}{2 π i ℏ (t^{''} - t^{'})})}^{1 / 2} \exp \frac{i m {(x^{''} - x^{'})}^{2}}{2 ℏ (t^{''} - t^{'})}

Note that one can derive all the harmonic-oscillator wave functions from the solution by use of the formula

⟨ x^{''}, t^{''}; x^{'}, t^{'} ⟩ = \sum_{n} u_{n} (x^{''}) u_{n}^{*} (x^{'}) \exp i E_{n} (t^{'} - t^{''}) / ℏ

Exercise 43.4. QUANTUM PROPAGATOR FOR FREE ELECTROMAGNETIC FIELD

In flat spacetime, one is given on the spacelike hypersurface

t = t^{'}

the divergence-free magnetic field

B^{'} (x, y, z)

and on the spacelike hypersurface

t = t^{''}

the divergence-free magnetic field

B^{''} (x, y, z)

. By Fourier analysis (reducing this problem to the preceding problem) or otherwise, find the probability amplitude to transit from

B^{'}

t^{'}

B^{''}

t^{''}

Exercise 43.5. HAMILTON-JACOBI FORMULATION OF MAXWELL ELECTRODYNAMICS

Regard the four components

A_{μ}

of the electromagnetic 4-potential as the primary quantities; split them into a space part

A_{i}

and a scalar potential

ϕ

. (1) Derive from the action principle (in flat spacetime)

I = (1 / 8 π) \int (E^{2} - B^{2}) d^{4} x

by splitting off an appropriate divergence, an expression qualitatively similar in character to (43.7). (2) Show that the appropriate quantity to be fixed on the initial and final spacelike hypersurface is not really

A_{i}

itself, but the magnetic field, defined by

B = curl A

. (3) Derive the Hamilton-Jacobi equation for the dynamic phase or action

S (B, S)

in its dependence on the choice of hypersurface

S

, and the choice of magnetic field

B

on this hypersurface,

- \frac{δ S}{δ Ω} = \frac{1}{8 π} B^{2} + \frac{(4 π)^{2}}{8 π} {(\frac{δ S}{δ A})}^{2}

The quantity on the left is Tomonaga's "bubble time" derivative [Tomonaga (1946); see also Box 21.1].

$- 44$

BEYOND THE END OF TIME

"Heaven wheels above youDisplaying to you her eternal glories And still your eyes are on the ground"

DANTE
The world "stands before us as a great, eternal riddle"
EINSTEIN (1949a)

This chapter is entirely Track 2. No previous Track-2 material is needed as preparation for it, but Chapter 43 will be helpful.

§44.1. GRAVITATIONAL COLLAPSE AS THE GREATEST CRISIS IN PHYSICS OF ALL TIME

The universe starts with a big bang, expands to a maximum dimension, then recontracts and collapses: no more awe-inspiring prediction was ever made. It is preposterous. Einstein himself could not believe his own prediction. It took Hubble's observations to force him and the scientific community to abandon the concept of a universe that endures from everlasting to everlasting.

Later work of Tolman (1934a), Avez (1960), Geroch (1967), and Hawking and Penrose (1969) generalizes the conclusion. A model universe that is closed, that obeys Einstein's geometrodynamic law, and that contains a nowhere negative density of mass-energy, inevitably develops a singularity. No one sees any escape from the density of mass-energy rising without limit. A computing machine calculating ahead step by step the dynamical evolution of the geometry comes to the point where it can not go on. Smoke, figuratively speaking, starts to pour out of the computer. Yet physics surely continues to go on if for no other reason than this: Physics is by definition that which does go on its eternal way despite all the shadowy changes in the surface appearance of reality.

Some day a door will surely open and expose the glittering central mechanism of the world in its beauty and simplicity. Toward the arrival of that day, no development holds out more hope than the paradox of gravitational collapse. Why paradox? Because Einstein's equation says "this is the end" and physics says "there is no end." Why hope? Because among all paradigms for probing a puzzle, physics proffers none with more promise than a paradox.

No previous period of physics brought a greater paradox than 1911 (Box 44.1). Rutherford had just been forced to conclude that matter is made up of localized positive and negative charges. Matter as so constituted should undergo electric collapse in

\sim 10^{- 17} \sec

, according to theory. Observation equally clearly proclaimed that matter is stable. No one took the paradox more seriously than Bohr. No one worked around and around the central mystery with more energy wherever work was possible. No one brought to bear a more judicious combination of daring and conservativeness, nor a deeper feel for the harmony of physics. The direct opposite

The paradox of collapse: physics stops, but physics must go on

The 1911 crisis of electric collapse

Box 44.1 COLLAPSE OF UNIVERSE PREDICTED BY CLASSICAL THEORY, COMPARED AND CONTRASTED WITH CLASSICALLY PREDICTED COLLAPSE OF ATOM

System

Atom (1911)

Universe (1970's)

Dynamic entity

System of electrons

Geometry of space

Nature of classi-

cally predicted

collapse

Electron headed toward point-center of

attraction is driven in a finite time

to infinite energy

Not only matter but space itself arrives

in a finite proper time at a condition

of infinite compaction

One rejected "way

out"

Give up Coulomb law of force

Give up Einstein's field equation

of harmony, cacophony, is the impression that comes from sampling the literature of the 'teens on the structure of the atom. (1) Change the Coulomb law of force between electric charges? (2) Give up the principle that an accelerated charge must radiate? There was little inhibition against this and still wilder abandonings of the well-established. In contrast, Bohr held fast to these two principles. At the same time he insisted on the importance of a third principle, firmly established by Planck in quite another domain of physics, the quantum principle. With that key he opened the door to the world of the atom.

Great as was the crisis of 1911, today gravitational collapse confronts physics with its greatest crisis ever. At issue is the fate, not of matter alone, but of the universe itself. The dynamics of collapse, or rather of its reverse, expansion, is evidenced not by theory alone, but also by observation; and not by one observation, but by observations many in number and carried out by astronomers of unsurpassed ability and integrity. Collapse, moreover, is not unique to the large-scale dynamics of the universe. A white dwarf star or a neutron star of more than critical mass is predicted to undergo gravitational collapse to a black hole (Chapters 32 and 33 ). Sufficiently many stars falling sufficiently close together at the center of the nucleus of a galaxy are expected to collapse to a black hole many powers of ten more massive than the sun. An active search is under way for evidence for a black hole in this Galaxy (Box 33.3). The process that makes a black hole is predicted to provide an experimental model for the gravitational collapse of the universe itself, with one difference. For collapse to a black hole, the observer has his choice whether (1) to observe from a safe distance, in which case his observations will reveal nothing of what goes on inside the horizon; or (2) to follow the falling matter on in, in which case he sees the final stages of the collapse, not only of the matter itself, but of the geometry surrounding the matter, to indefinitely high compaction, but only at the cost of his own early demise. For the gravitational collapse of a closed model universe, no such choice is available to the observer. His fate is sealed. So too is the fate of matter and elementary particles, driven up to indefinitely high densities. The stakes in the crisis of collapse are hard to match: the dynamics of the largest object, space, and the smallest object, an elementary particle-and how both began.

§44.2. ASSESSMENT OF THE THEORY THAT PREDICTS COLLAPSE

No one reflecting on the paradox of collapse ("collapse ends physics"; "collapse cannot end physics") can fail to ask, "What are the limits of validity of Einstein's geometric theory of gravity?" A similar question posed itself in the crisis of 1911. The Coulomb law for the force acting between two charges had been tested at distances of meters and millimeters, but what warrant was there to believe that it holds down to the

10^{- 8} cm

of atomic dimensions? Of course, in the end it proves to hold not only at the level of the atom, and at the

10^{- 13} cm

level of the nucleus, but even down to

5 \times 10^{- 15} cm

[Barber, Gittelman, O'Neill, and Richter, and Bailey et al. (1968), as reviewed by Farley (1969) and Brodsky and Drell (1970)], a striking
example of what Wigner (1960) calls the "unreasonable effectiveness of mathematics in the natural sciences."

No theory more resembles Maxwell's electrodynamics in its simplicity, beauty, and scope than Einstein's geometrodynamics. Few principles in physics are more firmly established than those on which it rests: the local validity of special relativity (Chapters 2-7), the equivalence principle (Chapter 16), the conservation of momentum and energy (Chapters 5, 15 and 16), and the prevalence of second-order field equations throughout physics (Chapter 17). Those principles and the demand for no "extraneous fields" (e.g., Dicke’s scalar field) and "no prior geometry" (§17.6) lead to the conclusion that the geometry of spacetime must be Riemannian and the geometrodynamic law must be Einstein's.

To say that the geometry is Riemannian is to say that the interval between any two nearby events

C

and

D

, anywhere in spacetime, stated in terms of the interval

A B

between two nearby fiducial events, at quite another point in spacetime, has a value

C D / A B

independent of the route of intercomparison (Chapter 13 and Box 16.4). There are a thousand routes. By this hydraheaded prediction, Einstein's theory thus exposes itself to destruction in a thousand ways (Box 16.4).

Geometrodynamics lends itself to being disproven in other ways as well. The geometry has no option about the control it exerts on the dynamics of particles and fields (Chapter 20). The theory makes predictions about the equilibrium configurations and pulsations of compact stars (Chapters 23-26). It gives formulas (Chapters 27-29) for the deceleration of the expansion of the universe, for the density of mass-energy, and for the magnifying power of the curvature of space, the tests of which are not far off. It predicts gravitational collapse, and the existence of black holes, and a wealth of physics associated with these objects (Chapters 31-34). It predicts gravitational waves (Chapters 35-37). In the appropriate approximation, it encompasses all the well-tested predictions of the Newtonian theory of gravity for the dynamics of the solar system, and predicts testable post-Newtonian corrections besides, including several already verified effects (Chapters 38-40).

No inconsistency of principle has ever been found in Einstein's geometric theory of gravity. No purported observational evidence against the theory has ever stood the test of time. No other acceptable account of physics of comparable simplicity and scope has ever been put forward.

Continue this assessment of general relativity a little further before returning to the central issue, the limits of validity of the theory and their bearing on the issue of gravitational collapse. What has Einstein's geometrodynamics contributed to the understanding of physics?
First, it has dethroned spacetime from a post of preordained perfection high above the battles of matter and energy, and marked it as a new dynamic entity participating actively in this combat.

Second, by tying energy and momentum to the curvature of spacetime, Einstein's theory has recognized the law of conservation of momentum and energy as an automatic consequence of the geometric identity that the boundary of a boundary is zero (Chapters 15 and 17).

Third, it has recognized gravitation as a manifestation of the curvature of the

New view of nature flowing
from Einstein's
geometrodynamics

Electric charge as lines of force trapped in the topology of space
geometry of spacetime rather than as something foreign and "physical" imbedded in spacetime.

Fourth, general relativity has reinforced the view that "physics is local"; that the analysis of physics becomes simple when it connects quantities at a given event with quantities at immediately adjacent events.

Fifth, obedient to the quantum principle, it recognizes that spacetime and time itself are ideas valid only at the classical level of approximation; that the proper arena for the Einstein dynamics of geometry is not spacetime, but superspace; and that this dynamics is described in accordance with the quantum principle by the propagation of a probability amplitude through superspace (Chapter 43). In consequence, the geometry of space is subject to quantum fluctuations in metric coefficients of the order

δ g \sim \frac{(Planck length, L^{*} = {(ℏ G / c^{3})}^{1 / 2} = 1.6 \times 10^{- 33} cm)}{(linear extension of region under study)}

Sixth, standard Einstein geometrodynamics is partial as little to Euclidean topology as to Euclidean geometry. A multiply connected topology provides a natural description for electric charge as electric lines of force trapped in the topology of a multiply connected space (Figure 44.1). Any other description of electricity postulates a breakdown in Maxwell's field equations for the vacuum at a site where charge

Figure 44.1.
Electric charge viewed as electric lines of force trapped in the topology of a multiply connected space [for the history of this concept see reference 36 of Wheeler (1968a)]. The wormhole or handle is envisaged as connecting two very different regions in the same space. One of the wormhole mouths, viewed by an observer with poor resolving power, appears to be the seat of an electric charge. Out of this region of 3 -space he finds lines of force emerging over the whole

4 π

solid angle. He may construct a boundary around this charge, determine the flux through this boundary, incorrectly apply the theorem of Gauss and "prove" that there is a charge "inside the boundary." It isn't a boundary. Someone caught within it-to speak figuratively-can go into that mouth of the wormhole, through the throat, out the other mouth, and return by way of the surrounding space to look at his "prison" from the outside. Lines of force nowhere end. Maxwell's equations nowhere fail. Nowhere can one place a finger and say, "Here there is some charge." This classical type of electric charge has no direct relation whatsoever to quantized electric charge. There is a freedom of choice about the flux through the wormhole, and a specificity about the connection between one charge and another, which is quite foreign to the charges of elementary particle physics. For ease of visualization the number of space dimensions in the above diagram has been reduced from three to two. The third dimension, measured off the surface, has no physical meaning-it only provides an extra dimension in which to imbed the surface for more convenient diagrammatic representation. [For more detail see Misner and Wheeler (1957), reprinted in Wheeler (1962)].
is located, or postulates the existence of some foreign and "physical" electric jelly imbedded in space, or both. No one has ever found a way to describe electricity free of these unhappy features except to say that the quantum fluctuations in the geometry of space are so great at small distances that even the topology fluctuates, makes "wormholes," and traps lines of force. These fluctuations have to be viewed, not as tied to particles, and endowed with the scale of distances associated with particle physics

(\sim 10^{- 13} cm)

but as pervading all space ("foam-like structure of geometry") and characterized by the Planck distance (

\sim 10^{- 33} cm

). Thus a third type of gravitational collapse forces itself on one's attention, a collapse continually being done and being undone everywhere in space: surely a guide to the outcome of collapse at the level of a star and at the level of the universe (Box 44.2).

Box 44.2 THREE LEVELS OF GRAVITATIONAL COLLAPSE

Universe
Black hole
Fluctuations at the Planck scale of distances

Recontraction and collapse of the universe is a kind of mirror image of the "big bang," on which one already has so much evidence.

Collapse of matter to form a black hole is most natural at two distinct levels: (a) collapse of the dense white-dwarf core of an individual star (when that core exceeds the critical mass,

\sim 1 M_{⊙}

2 M_{⊙}

, at which a neutron star is no longer a possible stable end-point for collapse) and (b) coalescence one by one of the stars in a galactic nucleus to make a black hole of mass up to

10^{6} M_{⊙}

or even

10^{9} M_{⊙}

In either case, no feature of principle about matter falling into the black hole is more interesting than the option that the observer has (symbolized by the branching arrow in the inset). He can go along with the infalling matter, in which case he sees the final stages of collapse, but only at the cost of his own demise. Or he can stay safely outside, in which case even after in-

definitely long time he sees only the first part of the collapse, with the infalling matter creeping up more and more slowly to the horizon.

In the final stages of the collapse of a closed model universe, all black holes present are caught up and driven together, amalgamating one by one. No one has any way to look at the event from safely outside; one is inevitably caught up in it oneself.

Collapse at the Planck scale of distances is taking place everywhere and all the time in quantum fluctuations in the geometry and, one believes, the topology of space. In this sense, collapse is continually being done and undone, modeling the undoing of the collapse of the universe itself, summarized in the term, "the reprocessing of the universe" (see text).

§44.3. VACUUM FLUCTUATIONS: THEIR PREVALENCE AND FINAL DOMINANCE

Is matter built out of geometry?

The richness of the physics of the vacuum

If Einstein's theory thus throws light on the rest of physics, the rest of physics also throws light on geometrodynamics. No point is more central than this, that empty space is not empty. It is the seat of the most violent physics. The electromagnetic field fluctuates (Chapter 43). Virtual pairs of positive and negative electrons, in effect, are continually being created and annihilated, and likewise pairs of mu mesons, pairs of baryons, and pairs of other particles. All these fluctuations coexist with the quantum fluctuations in the geometry and topology of space. Are they additional to those geometrodynamic zero-point disturbances, or are they, in some sense not now well-understood, mere manifestations of them?

Put the question in other words. Recall Clifford, inspired by Riemann, speaking to the Cambridge Philosophical Society on February 21, 1870, "On the Space Theory of Matter" [Clifford (1879), pp. 244 and 322; and (1882), p. 21], and saying, "I hold in fact (1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them. (2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave. (3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial. (4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity." Ask if there is a sense in which one can speak of a particle as constructed out of geometry. Or rephrase the question in updated language: "Is a particle a geometrodynamic exciton?" What else is there out of which to build a particle except geometry itself? And what else is there to give discreteness to such an object except the quantum principle?

The Clifford-Einstein space theory of matter has not been forgotten in recent years. "In conclusion," one of the authors wrote a decade ago [Wheeler (1962)], "the vision of Riemann, Clifford, and Einstein, of a purely geometric basis for physics, today has come to a higher state of development, and offers richer prospects-and presents deeper problems-than ever before. The quantum of action adds to this geometrodynamics new features, of which the most striking is the presence of fluctuations of the wormhole type throughout all space. If there is any correspondence at all between this virtual foam-like structure and the physical vacuum as it has come to be known through quantum electrodynamics, then there seems to be no escape from identifying these wormholes with 'undressed electrons.' Completely different from these 'undressed electrons,' according to all available evidence, are the electrons and other particles of experimental physics. For these particles the geometrodynamic picture suggests the model of collective disturbances in a virtual foam-like vacuum, analogous to different kinds of phonons or excitons in a solid.
"The enormous factor from nuclear densities

\sim 10^{14} g / {cm}^{3}

to the density of field fluctuation energy in the vacuum,

\sim 10^{94} g / {cm}^{3}

, argues that elementary particles represent a percentage-wise almost completely negligible change in the locally violent conditions that characterize the vacuum. ['A particle

(10^{14} g / {cm}^{3})

means as little
to the physics of the vacuum

(10^{94} g / {cm}^{3})

as a cloud

(10^{- 6} g / {cm}^{3})

means to the physics of the sky

(10^{- 3} g / {cm}^{3}) .^{'}]

In other words, elementary particles do not form a really basic starting point for the description of nature. Instead, they represent a first-order correction to vacuum physics. That vacuum, that zero-order state of affairs, with its enormous densities of virtual photons and virtual positive-negative pairs and virtual wormholes, has to be described properly before one has a fundamental starting point for a proper perturbation-theoretic analysis."
"These conclusions about the energy density of the vacuum, its complicated topological character, and the richness of the physics which goes on in the vacuum, stand in no evident contradiction with what quantum electrodynamics has to say about the vacuum. Instead the conclusions from the 'small distance' analysis

(10^{- 33} cm)

-sketchy as it is-and from 'larger distance' analysis

(10^{- 11} cm)

would seem to [be able] to reinforce each other in a most natural way.
"The most evident shortcoming of the geometrodynamic model as it stands is this, that it fails to supply any completely natural place for spin

\frac{1}{2}

in general and for the neutrino in particular."

Attempts to find a natural place for spin

\frac{1}{2}

in Einstein's standard geometrodynamics (Box 44.3) founder because there is no natural way for a change in connectivity to take place within the context of classical differential geometry.

A uranium nucleus undergoing fission starts with one topology and nevertheless ends up with another topology. It makes this transition in a perfectly continuous way, classical differential geometry notwithstanding.

There are reputed to be two kinds of lawyers. One tells the client what not to do. The other listens to what the client has to do and tells him how to do it. From the first lawyer, classical differential geometry, the client goes away disappointed, still searching for a natural way to describe quantum fluctuations in the connectivity of space. Only in this way can he hope to describe electric charge as lines of electric force trapped in the topology of space. Only in this way does he expect to be able to understand and analyze the final stages of gravitational collapse. Pondering his problems, he comes to the office of a second lawyer, with the name "Pregeometry" on the door. Full of hope, he knocks and enters. What is pregeometry to be and say? Born of a combination of hope and need, of philosophy and physics and mathematics and logic, pregeometry will tell a story unfinished at this writing, but full of incidents of evolution so far as it goes.

§44.4. NOT GEOMETRY, BUT PREGEOMETRY, AS THE MAGIC BUILDING MATERIAL

An early survey (Box 44.4) asked whether geometry can be constructed with the help of the quantum principle out of more basic elements, that do not themselves have any specific dimensionality.

The focus of attention in this 1964 discussion was "dimensionality without dimensionality." However, the prime pressures to ponder pregeometry were and remain

No place in geometrodynamics for change of topology; therefore turn to "pregeometry"

Box 44.3 THE DIFFICULTIES WITH ATTEMPTS TO FIND A NATURAL PLACE FOR SPIN $\frac{1}{2}$ IN EINSTEIN'S STANDARD GEOMETRODYNAMICS

"It is impossible" [Wheeler (1962)] "to accept any description of elementary particles that does not have a place for

spin \frac{1}{2}

. What, then, has any purely geometric description to offer in explanation of

spin \frac{1}{2}

in general? More particularly and more importantly, what place is there in quantum geometrodynamics for the neutrino-the only entity of half-integral spin that is a pure field in its own right, in the sense that it has zero rest mass and moves with the speed of light? No clear or satisfactory answer is known to this question today. Unless and until an answer is forthcoming, pure geometrodynamics must be judged deficient as a basis for elementary particle physics."

A later publication [Wheeler (1968a)] takes up this issue again, noting that, "A new world opens out for analysis in quantum geometrodynamics. The central new concept is space resonating between one foamlike structure and another. For this multiple-connectedness of space at submicroscopic distances no single feature of nature speaks more powerfully than electric charge. Yet at least as impressive as charge is the prevalence of spin

\frac{1}{2}

throughout the world of elementary particles."

Repeating the statement that "It is impossible to accept any description of elementary particles that does not have a place for spin

\frac{1}{2}

," the article adds to the discussion a new note: "Happily, the concept of spin manifold has come to light, not least through the work of John Milnor [see Lichnerowicz (1961a,b,c) and (1964); Milnor (1962), (1963), and (1965a,b); Hsiang and Anderson (1965); Anderson, Brown, and Peterson (1966a,b); and Penrose (1968a)]. This concept suggests a new and interesting interpretation of a spinor field within the context of the resonating microtopology of quantum geometrodynamics, as the nonclassical two-valuedness [Pauli's standard term for spin; see, for example, Pauli (1947)] that attaches to the probability amplitude for otherwise identical 3-geometries endowed with alternative 'spin structures.'" More specifically: "One does not classify the closed orientable 3-manifold of physics completely
when one gives its topology, its differential structures, and its metric. One must tell which spin structure it has." [On a 3-geometry with the topology of a 3 -sphere, one can lay down a continuous field of triads (a triad consisting of three orthonormal vectors). Any other continuous field of triads can be deformed into the first field by a continuous sequence of small readjustments. One says that the 3 -sphere admits only one "spin structure," a potentially misleading standard word for what could just as well have been called a "triad structure." In contrast, a 3 -sphere with

n

handles or wormholes admits

2^{n}

"spin structures" (continuous fields of triads) inequivalent to one another under any continuous sequence of small readjustments whatsoever, and distinguished from one another in any convenient way by

n

"descriptors"

w_{1}, w_{2}, \dots

w_{k}, \dots w_{n}

. I It is natural in quantum geometrodynamics to expect "separate probability amplitudes for a 3-geometry with descriptor

w_{k} = + 1

and for an otherwise identical 3 -geometry with descriptor

w_{k} = - 1

. Does this circumstance imply that quantum geometrodynamics supplies all the machinery one needs to describe fields of spin

\frac{1}{2}

in general and the neutrino field in particular? . . . That is the only way that has ever turned up within the framework of Einstein's general relativity and Planck's quantum principle. Is this the right path? It is difficult to name any question more decisive than this in one's assessment of 'everything as geometry.'"

Why not spell out these concepts, reduce them to practice, and compare them with what one knows about the behavior of fields of spin

\frac{1}{2}

? There is a central difficulty in this enterprise. It assumes and demands on physical grounds that the topology of the 3-geometry shall be free to change from one connectivity to another. In contrast, classical differential geometry says, in effect, "Once one topology, always that topology." Try a question like this, "When a new handle develops and the number of descriptors rises by one, what boundary condition in superspace connects the probability
amplitude

ψ

for 3-geometries of the original topology with the probability amplitudes

ψ_{+}

and

ψ_{-}

for the two spin structures of the new topology?" Classical differential geometry not only gives one no help in answering this question; it even forbids one to ask it. In other words, one cannot even get the enterprise "on the road" for want of a natural
mathematical way to describe the required change in topology. The idea is therefore abandoned here and now that 3 -geometry is "the magic building material of the universe." In contrast, pregeometry (see text), far from being endowed with any frozen topology, is to be viewed as not even possessing any dimensionality.

Box 44.4 "A BUCKET OF DUST"-AN EARLY ATTEMPT TO FORMULATE THE CONCEPT OF PREGEOMETRY [Wheeler (1964a)]

"What line of thought could ever be imagined as leading to four dimensions-or any dimensionality at all-out of more primitive considerations? In the case of atoms one derives the yellow color of the sodium D-lines by analyzing the quantum dynamics of a system, no part of which is ever endowed with anything remotely resembling the attribute of color. Likewise any derivation of the four-dimensionality of spacetime can hardly start with the idea of dimensionality."
"Recall the notion of a Borel set. Loosely speaking, a Borel set is a collection of points ("bucket of dust") which have not yet been assembled into a manifold of any particular dimensionality. . . . Recalling the universal sway of the quantum principle, one can imagine probability amplitudes for the points in a Borel set to be assembled into points with this, that, and the other dimensionality. . . . More conditions have to be imposed on a given number of points-as to which has which for a nearest neighbor-when the points are put together in a five-dimensional array than when these same points are arranged in a two-dimensional pattern. Thus one can think of each dimensionality as having a much higher statistical weight than the next higher dimensionality. On the other hand, for manifolds with one, two, and three dimensions, the geometry is too rudimentary-one can suppose-to give anything interesting. Thus Einstein's field equations, applied to a manifold of dimensionality so low, demand flat space; only when the dimensionality is as high as four do really interesting possibilities arise. Can four,
therefore, be considered to be the unique dimensionality which is at the same time high enough to give any real physics and yet low enough to have great statistical weight?
"It is too much to imagine that one has yet made enough mistakes in this domain of thought to explore such ideas with any degree of good judgment."

Consider a handle on the geometry. Let it thin halfway along its length to a point. In other words, let the handle dissolve into two bent prongs that touch at a point. Let these prongs separate and shorten. In this process two points part company that were once immediate neighbors. "However sudden the change is in classical theory, in quantum theory there is a probability amplitude function which falls off in the classically forbidden domain. In other words, there is some residual connection between points which are ostensibly very far away (travel from one 'tip' down one prong, then through the larger space to which these prongs are attached, and then up the other prong to the other tip). But there is nothing distinctive in principle about the two points that have happened to come into discussion. Thus it would seem that there must be a connection . . . between every point and every other point. Under these conditions the concept of nearest neighbor would appear no longer to make sense. Thus the tool disappears with the help of which one might otherwise try to speak [un]ambiguously about dimensionality."

Sakharov: gravitation is the "metric elasticity of space"

The stratification of space

Comparison with everyday elasticity
two features of nature, spin

\frac{1}{2}

and charge, that speak out powerfully from every part of elementary particle physics.

A fresh perspective on pregeometry comes from a fresh assessment of general relativity. "Geometrodynamics is neither as important or as simple as it looks. Do not make it the point of departure in searching for underlying simplicity. Look deeper, at elementary particle physics." This is the tenor of interesting new considerations put forward by Sakharov [the Sakharov] (1967) and summarized under the heading, "Gravitation as the metric elasticity of space," in Box 17.2. In brief, as elasticity is to atomic physics, so-in Sakharov's view-gravitation is to elementary particle physics. The energy of an elastic deformation is nothing but energy put into the bonds between atom and atom by the deformation. The energy that it takes to curve space is nothing but perturbation in the vacuum energy of fields plus particles brought about by that curvature, according to Sakharov. The energy required for the deformation is governed in the one case by two elastic constants and in the other case by one elastic constant (the Newtonian constant of gravity) but in both cases, he reasons, the constants arise by combination of a multitude of complicated individual effects, not by a brave clean stroke on an empty slate.

One gives all the more favorable reception to Sakhorov's view of gravity because one knows today, as one did not in 1915, how opulent in physics the vacuum is. In Einstein's day one had come in a single decade from the ideal God-given Lorentz perfection of flat spacetime to curved spacetime. It took courage to assign even one physical constant to that world of geometry that had always stood so far above physics. The vacuum looked for long as innocent of structure as a sheet of glass emerging from a rolling mill. With the discovery of the positive electron [Anderson (1933)], one came to recognize a little of the life that heat can unfreeze in "empty" space. Each new particle and radiation that was discovered brought a new accretion to the recognized richness of the vacuum. Macadam looks smooth, but a bulldozer has only to cut a single furrow through the roadway to disclose all the complications beneath the surface.

Think of a particle as built out of the geometry of space; think of a particle as a "geometrodynamic exciton"? No model-it would seem to follow from Sakharov's assessment-could be less in harmony with nature, except to think of an atom as built out of elasticity! Elasticity did not explain atoms. Atoms explained elasticity. If, likewise, particles fix the constant in Einstein's geometrodynamic law (Sakharov), must it not be unreasonable to think of the geometrodynamic law as explaining particles?

Carry the comparison between geometry and elasticity one stage deeper (Fig. 44.2). In a mixed solid there are hundreds of distinct bonds, all of which contribute to the elastic constants; some of them arise from Van der Waal's forces, some from ionic coupling, some from homopolar linkage; they have the greatest variety of strengths; but all have their origin in something so fantastically simple as a system of positively and negatively charged masses moving in accordance with the laws of quantum mechanics. In no way was it required or right to meet each complication of the chemistry and physics of a myriad of bonds with a corresponding complication of principle. By going to a level of analysis deeper than bond strengths, one had

Figure 44.2.
Elasticity and geometrodynamics, as viewed at three levels of analysis. A hundred years of the study of elasticity did not reveal the existence of molecules, and a hundred years of the study of molecular chemistry did not reveal Schrödinger's equation. Revelation moved upward in the diagram, not downward.
emerged into a world of light, where nothing but simplicity and unity was to be seen.
Compare with geometry. The vacuum is animated with the zero-point activity of distinct fields and scores of distinct particles, all of which, according to Sakharov, contribute to the Newtonian

G

, the "elastic constant of the metric." Some interact via weak forces, some by way of electromagnetic forces, and some through strong forces. These interactions have the greatest variety of strengths. But must not all these particles and interactions have their origin in something fantastically simple? And must not this something, this "pregeometry," be as far removed from geometry as the quantum mechanics of electrons is far removed from elasticity?
If one once thought of general relativity as a guide to the discovery of pregeometry, nothing might seem more dismaying than this comparison with an older realm of physics. No one would dream of studying the laws of elasticity to uncover the principles of quantum mechanics. Neither would anyone investigate the work-hardening of a metal to learn about atomic physics. The order of understanding ran not

Work-hardening (1 cm) ⟶ dislocations (10^{- 4} cm) ⟶ atoms (10^{- 8} cm),

but the direct opposite,

Atoms (10^{- 8} cm) ⟶ dislocations (10^{- 4} cm) ⟶ work-hardening (1 cm)

One had to know about atoms to conceive of dislocations, and had to know about dislocations to understand work-hardening. Is it not likewise hopeless to go from the "elasticity of geometry" to an understanding of particle physics, and from particle physics to the uncovering of pregeometry? Must not the order of progress again be the direct opposite? And is not the source of any dismay the apparent loss of guidance that one experiences in giving up geometrodynamics-and not only geometrodynamics, but geometry itself-as a crutch to lean on as one hobbles forward? Yet there is so much chance that this view of nature is right that one must take it seriously and explore its consequences. Never more than today does one have the incentive to explore pregeometry.

§44.5. PREGEOMETRY AS THE CALCULUS OF PROPOSITIONS

Paper in white the floor of the room, and rule it off in one-foot squares. Down on one's hands and knees, write in the first square a set of equations conceived as able to govern the physics of the universe. Think more overnight. Next day put a better set of equations into square two. Invite one's most respected colleagues to contribute to other squares. At the end of these labors, one has worked oneself out into the door way. Stand up, look back on all those equations, some perhaps more hopeful than others, raise one's finger commandingly, and give the order "Fly!" Not one of those equations will put on wings, take off, or fly. Yet the universe "flies."

Some principle uniquely right and uniquely simple must, when one knows it, be also so compelling that it is clear the universe is built, and must be built, in such and such a way, and that it could not possibly be otherwise. But how can one discover that principle? If it was hopeless to learn atomic physics by studying work-hardening and dislocations, it may be equally hopeless to learn the basic operating principle of the universe, call it pregeometry or call it what one will, by any amount of work in general relativity and particle physics.

Thomas Mann (1937), in his essay on Freud, utters what Niels Bohr would surely have called a great truth ("A great truth is a truth whose opposite is also a great truth") when he says, "Science never makes an advance until philosophy authorizes and encourages it to do so." If the equivalence principle (Chapter 16) and Mach's principle (

$ 21.9

) were the philosophical godfathers of general relativity, it is also true that what those principles do mean, and ought to mean, only becomes clear by study and restudy of Einstein's theory itself. Therefore it would seem reasonable to expect the primary guidance in the search for pregeometry to come from a principle both philosophical and powerful, but one also perhaps not destined to be wholly clear in its contents or its implications until some later day.

Among all the principles that one can name out of the world of science, it is difficult to think of one more compelling than simplicity; and among all the simplicities of dynamics and life and movement, none is starker [Werner (1969)] than the binary choice yes-no or true-false. It in no way proves that this choice for a starting principle is correct, but it at least gives one some comfort in the choice, that Pauli's "nonclassical two-valuedness" or "spin" so dominates the world of particle physics.

It is one thing to have a start, a tentative construction of pregeometry; but how does one go on? How not to go on is illustrated by Figure 44.3. The "sewing machine" builds objects of one or another definite dimensionality, or of mixed dimensionalities, according to the instructions that it receives on the input tape in yes-no binary code. Some of the difficulties of building up structure on the binary element according to this model, or any one of a dozen other models, stand out at once. (1) Why

N = 10, 000

building units? Why not a different

N

? And if one feeds in one such arbitrary number at the start, why not fix more features "by hand?" No natural stopping point is evident, nor any principle that would fix such a stopping point. Such arbitrariness contradicts the principle of simplicity and rules out the model. (2) Quantum mechanics is added from outside, not generated from inside (from the model itself). On this point too the principle of simplicity speaks against the model. (3) The passage from pregeometry to geometry is made in a too-literal-minded way, with no appreciation of the need for particles and fields to appear along the way. The model, in the words used by Bohr on another occasion, is "crazy, but not crazy enough to be right."

Noting these difficulties, and fruitlessly trying model after model of pregeometry to see if it might be free of them, one suddenly realizes that a machinery for the combination of yes-no or true-false elements does not have to be invented. It already exists. What else can pregeometry be, one asks oneself, than the calculus of propositions? (Box 44.5.)

§44.6. THE BLACK BOX: THE REPROCESSING OF THE UNIVERSE

No amount of searching has ever disclosed a "cheap way" out of gravitational collapse, any more than earlier it revealed a cheap way out of the collapse of the atom. Physicists in that earlier crisis found themselves in the end confronted with a revolutionary pistol, "Understand nothing-or accept the quantum principle." Today's crisis can hardly force a lesser revolution. One sees no alternative except to say that geometry fails and pregeometry has to take its place to ferry physics through the final stages of gravitational collapse and on into what happens next. No guide is evident on this uncharted way except the principle of simplicity, applied to drastic lengths.

Whether the whole universe is squeezed down to the Planck dimension, or more or less, before reexpansion can begin and dynamics can return to normal, may be irrelevant for some of the questions one wants to consider. Physics has long used the "black box" to symbolize situations where one wishes to concentrate on what goes in and what goes out, disregarding what takes place in between.

At the beginning of the crisis of electric collapse one conceived of the electron as headed on a deterministic path toward a point-center of attraction, and unhappily destined to arrive at a condition of infinite kinetic energy in a finite time. After the advent of quantum mechanics, one learned to summarize the interaction between

A first try at a pregeometry built on the principle of binary choice

A more reasonable picture: pregeometry is the calculus of propositions

The role of the black box in physics

Figure 44.3.
"Ten thousand rings"; or an example of a way to think of the connection between pregeometry and geometry, wrong because it is too literal-minded, and for other reasons spelled out in the text. The vizier [story by Wheeler, as alluded to by Kilmister (1971)*] speaks: "Take

N = 10, 000

brass rings. Take an automatic fastening device that will cut open a ring, loop it through another ring, and resolder the joint. Pour the brass rings into the hopper that feeds this machine. Take a strip of instruction paper that is long enough to contain

N (N - 1) / 2

binary digits. Look at the instruction in the

(j k)

-th location on this instruction tape

(j, k = 1, 2, \dots, N; j < k)

. When the binary digit at that location is 0 , it is a signal to leave the

j

-th ring disconnected from the

k

-th ring. When it is 1 , it is an instruction to connect that particular pair of rings. Thread the tape into the machine and press the start button. The clatter begins. Out comes a chain of rings 10,000 links long. It falls on the table and the machine stops. Pour in another 10,000 rings, feed in a new instruction tape, and push the button again. This time it is not a one-dimensional structure that emerges, but a two-dimensional one: a Crusader's coat of mail, complete with neck opening and sleeves. Take still another tape from the library of tapes and repeat. Onto the table thuds a smaller version of the suit of mail, this time filled out internally with a solid network of rings, a three-dimensional structure. Now forego the library and make one's own instruction tape, a random string of 0's and l's. Guided by it, the fastener produces a "Christmas tree ornament," a collection of segments of one-dimensional chain, two-dimensional surfaces, and three-, four-, five-, and higher-dimensional entities, some joined together, some free-floating. Now turn from a structure deterministically fixed by a tape to a probability amplitude, a complex number,

\begin{matrix} (1) & ψ (tape) = ψ (n_{12}, n_{13}, n_{14}, \dots \dots, n_{N - 1, N}) (n_{i j} = 0, 1) \end{matrix}

defined over the entire range of possibilities for structures built of 10,000 rings. Let these probability amplitudes not be assigned randomly. Instead, couple together amplitudes, for structures that differ from each other by the breaking of a single ring, by linear formulas that treat all rings on the same footing. The separate

ψ

's, no longer entirely independent, will still give non-zero probability amplitudes for "Christmas tree ornaments." Of greater immediate interest than these "unruly" parts of the structures are the following questions about the smoother parts: (1) In what kinds of structures is the bulk of the probability concentrated? (2) What is the dominant dimensionality of these structures in an appropriate correspondence principle limit? (3) In this semiclassical limit, what is the form taken by the dynamic law of evolution of the geometry?" No principle more clearly rules out this model for pregeometry than the principle of simplicity (see text).

Box 44.5 "PREGEOMETRY AS THE CALCULUS OF PROPOSITIONS"

A sample proposition taken out of a standard text on logic selected almost at random reads [Kneebone (1963), p. 40]

\begin{aligned} [X ⟶ & ((X ⟶ X) ⟶ Y)] & (\bar{X} ⟶ Z) eq (\bar{X} \lor Y \lor Z) & \\ (\bar{X} \lor Y \lor \bar{Z}) & (X \lor Y \lor Z) & (X \lor \bar{Y} \lor Z) . \end{aligned}

The symbols have the following meaning:

$\bar{A}$ ,	Not $A;$
$A \lor B$ ,	$A$ or $B$ or both (" $A$ vel $B$ ");
$A & B$ ,	$A$ and $B;$
$A ⟶ B$ ,	$A$ implies $B$ ("if $A$ , then $B$ ");
$A ⟷ B$ ,	$B$ is equivalent to $A$ (" $B$ if and only if $A$ ").

A ⟷

is a tautology. The letters

A, B

, etc., serve as connectors to "wire together" one proposition with another. Proceeding in this way, one can construct propositions of indefinitely great length.

A switching circuit [see, for example, Shannon (1938) or Hohn (1966)] is isomorphic to a proposition.

Compare a short proposition or an elementary switching circuit to a molecular collision. No idea seemed more preposterous than that of Daniel Bernoulli (1733), that heat is a manifestation of molecular collisions. Moreover, a three-body encounter is difficult to treat, a four-body collision is more difficult, and a five- or more molecule system is essentially intractable. Nevertheless, mechanics acquires new elements of simplicity in the limit in which the number of molecules is very great and in which one can use the concept of density in phase space. Out of statistical mechanics in this limit come such concepts as temperature and entropy. When the temperature is well-defined, the energy of the system is not a well-defined idea; and when the energy is well-defined, the temperature is not. This complementarity is built inescapably into the principles of the subject. Thrust the finger into the flame of a match and experience a sensation like nothing else on heaven or earth; yet what happens is all a consequence of molecular collisions, early critics notwithstanding.

Any individual proposition is difficult for the mind to apprehend when it is long; and still more difficult to grasp is the content of a cluster of propositions. Nevertheless, make a statistical analysis of the calculus of propositions in the limit where the number of propositions is great and most of them are long. Ask if parameters force themselves on one's attention in this analysis (1) analogous in some small measure to the temperature and entropy of statistical mechanics but (2) so much

Box 44.5 (continued)

more numerous, and everyday dynamic in character, that they reproduce the continuum of everyday physics.

Nothing could seem so preposterous at first glance as the thought that nature is built on a foundation as ethereal as the calculus of propositions. Yet, beyond the push to look in this direction provided by the principle of simplicity, there are two pulls. First, bare-bones quantum mechanics lends itself in a marvelously natural way to formulation in the language of the calculus of propositions, as witnesses not least the book of Jauch (1968). If the quantum principle were not in this way already automatically contained in one's proposed model for pregeometry, and if in contrast it had to be introduced from outside, by that very token one would conclude that the model violated the principle of simplicity, and would have to reject it. Second, the pursuit of reality seems always to take one away from reality. Who would have imagined describing something so much a part of the here and now as gravitation in terms of curvature of the geometry of spacetime? And when later this geometry came to be recognized as dynamic, who would have dreamed that geometrodynamics unfolds in an arena so ethereal as superspace? Little astonishment there should be, therefore, if the description of nature carries one in the end to logic, the ethereal eyrie at the center of mathematics. If, as one believes, all mathematics reduces to the mathematics of logic, and all physics reduces to mathematics, what alternative is there but for all physics to reduce to the mathematics of logic? Logic is the only branch of mathematics that can "think about itself."
"An issue of logic having nothing to do with physics" was the assessment by many of a controversy of old about the axiom, "parallel lines never meet." Does it follow from the other axioms of Euclidean geometry or is it independent? "Independent," Bolyai and Lobachevsky proved. With this and the work of Gauss as a start, Riemann went on to create Riemannian geometry. Study nature, not Euclid, to find out about geometry, he advised; and Einstein went on to take that advice and to make geometry a part of physics.
"An issue of logic having nothing to do with physics" is one's natural first assessment of the startling limitations on logic discovered by Gödel (1931), Cohen (1966), and others [for a review, see, for example, Kac and Ulam (1968)]. The exact opposite must be one's assessment if the real pregeometry of the real physical world indeed turns out to be identical with the calculus of propositions.
"Physics as manifestation of logic" or "pregeometry as the calculus of propositions" is as yet [Wheeler (1971a)] not an idea, but an idea for an idea. It is put forward here only to make it a little clearer what it means to suggest that the order of progress may not be

physics ⟶ pregeometry

but

pregeometry ⟶ physics.

Figure 44.4.
The "black-box model" applied (1) to the scattering of an electron by a center of attraction and (2) to the collapse of the universe itself. The deterministic electron world line of classical theory is replaced in quantum theory by a probability amplitude, the wave crests of which are illustrated schematically in the diagram. The catastrophe of classical theory is replaced in quantum theory by a probability distribution of outputs. The same diagram illustrates the "black-box account" of gravitational collapse mentioned in the text. The arena of the diagram is no longer spacetime, but superspace. The incident arrow marks no longer a classical world line of an electron through spacetime, but a classical "leaf of history of geometry" slicing through superspace (Chapter 43). The wave crests symbolize no longer the electron wave function propagating through spacetime, but the geometrodynamic wave function propagating through superspace. The cross-hatched region is no longer the region where the one-body potential goes to infinity, but the region of gravitational collapse where the curvature of space goes to infinity. The outgoing waves describe no longer alternative directions for the new course of the scattered electron, but the beginnings of alternative new histories for the universe itself after collapse and "reprocessing" end the present cycle.
center of attraction and electron in a "black box:" fire in a wave-train of electrons traveling in one direction, and get electrons coming out in this, that, and the other direction with this, that, and the other well-determined probability amplitude (Figure 44.4). Moreover, to predict these probability amplitudes quantitatively and correctly, it was enough to translate the Hamiltonian of classical theory into the language of wave mechanics and solve the resulting wave equation, the key to the "black box."

A similar "black box" view of gravitational collapse leads one to expect a "probability distribution of outcomes." Here, however, one outcome is distinguished from another, one must anticipate, not by a single parameter, such as the angle of scattering of the electron, but by many. They govern, one foresees, such quantities as the size of the system at its maximum of expansion, the time from the start of this new cycle to the moment it ends in collapse, the number of particles present, and a thousand other features. The "probabilities" of these outcomes will be governed by a dynamic law, analogous to (1) the Schrödinger wave equation for the electron, or, to cite another black box problem, (2) the Maxwell equations that couple together, at a wave-guide junction, electromagnetic waves running in otherwise separate wave guides. However, it is hardly reasonable to expect the necessary dynamic law to spring forth as soon as one translates the Hamilton-Jacobi equation of general relativity (Chapter 43) into a Schrödinger equation, simply because geometrodynamics, in both its classical and its quantum version, is built on standard differential geometry. That standard geometry leaves no room for any of those quantum fluctuations in connectivity that seem inescapable at small distances and therefore also inescapable in the final stages of gravitational collapse. Not geometry, but pregeometry, must fill the black box of gravitational collapse.

Probability distribution of the outcomes of collapse
"Reprocessing" the universe

All conservation laws transcended in the collapse of the universe

Little as one knows the internal machinery of the black box, one sees no escape from this picture of what goes on: the universe transforms, or transmutes, or transits, or is reprocessed probabilistically from one cycle of history to another in the era of collapse.

However straightforwardly and inescapably this picture of the reprocessing of the universe would seem to follow from the leading features of general relativity and the quantum principle, the two overarching principles of twentieth-century physics, it is nevertheless fantastic to contemplate. How can the dynamics of a system so incredibly gigantic be switched, and switched at the whim of probability, from one cycle that has lasted

10^{11}

years to one that will last only

10^{6}

years? At first, only the circumstance that the system gets squeezed down in the course of this dynamics to incredibly small distances reconciles one to a transformation otherwise so unbelievable. Then one looks at the upended strata of a mountain slope, or a bird not seen before, and marvels that the whole universe is incredible:

mutation of a species, metamorphosis of a rock, chemical transformation, spontaneous transformation of a nucleus, radioactive decay of a particle, reprocessing of the universe itself.

If it cast a new light on geology to know that rocks can be raised and lowered thousands of meters and hundreds of degrees, what does it mean for physics to think of the universe as being from time to time "squeezed through a knothole," drastically "reprocessed," and started out on a fresh dynamic cycle? Three considerations above all press themselves on one's attention, prefigured in these compressed phrases:

destruction of all constants of motion in collapse; particles, and the physical "constants" themselves, as the
"frozen-in part of the meteorology of collapse;"
"the biological selection of physical constants."

The gravitational collapse of a star, or a collection of stars, to a black hole extinguishes all details of the system (see Chapters 32 and 33 ) except mass and charge and angular momentum. Whether made of matter or antimatter or radiation, whether endowed with much entropy or little entropy, whether in smooth motion or chaotic turbulence, the collapsing system ends up as seen from outside, according to all indications, in the same standard state. The laws of conservation of baryon number and lepton number are transcended [Chapter 33; also Wheeler (1971b)]. No known means whatsoever will distinguish between black holes of the most different provenance if only they have the same mass, charge, and angular momentum. But for a closed universe, even these constants vanish from the scene. Total charge is automatically zero because lines of force have nowhere to end except upon charge. Total mass and total angular momentum have absolutely no definable meaning whatsoever for a closed universe. This conclusion follows not least because there
is no asymptotically flat space outside where one can put a test particle into Keplerian orbit to determine period and precession.

Of all principles of physics, the laws of conservation of charge, lepton number, baryon number, mass, and angular momentum are among the most firmly established. Yet with gravitational collapse the content of these conservation laws also collapses. The established is disestablished. No determinant of motion does one see left that could continue unchanged in value from cycle to cycle of the universe. Moreover, if particles are dynamic in construction, and if the spectrum of particle masses is therefore dynamic in origin, no option would seem left except to conclude that the mass spectrum is itself reprocessed at the time when "the universe is squeezed through a knot hole." A molecule in this piece of paper is a "fossil" from photochemical synthesis in a tree a few years ago. A nucleus of the oxygen in this air is a fossil from thermonuclear combustion at a much higher temperature in a star a few

10^{9}

years ago. What else can a particle be but a fossil from the most violent event of all, gravitational collapse?

That one geological stratum has one many-miles long slope, with marvelous linearity of structure, and another stratum has another slope, is either an everyday triteness, taken as for granted by every passerby, or a miracle, until one understands the mechanism. That an electron here has the same mass as an electron there is also a triviality or a miracle. It is a triviality in quantum electrodynamics because it is assumed rather than derived. However, it is a miracle on any view that regards the universe as being from time to time "reprocessed." How can electrons at different times and places in the present cycle of the universe have the same mass if the spectrum of particle masses differs between one cycle of the universe and another? Inspect the interior of a particle of one type, and magnify it up enormously, and in that interior see one view of the whole universe [compare the concept of monad of Leibniz (1714), "The monads have no window through which anything can enter or depart"]; and do likewise for another particle of the same type. Are particles of the same pattern identical in any one cycle of the universe because they give identically patterned views of the same universe? No acceptable explanation for the miraculous identity of particles of the same type has ever been put forward. That identity must be regarded, not as a triviality, but as a central mystery of physics.

Not the spectrum of particle masses alone, but the physical "constants" themselves, would seem most reasonably regarded as reprocessed from one cycle to another. Reprocessed relative to what? Relative, for example, to the Planck system of units,

\begin{aligned} L^{*} & = {(ℏ G / c^{3})}^{1 / 2} = 1.6 \times 10^{- 33} cm, \\ T^{*} & = {(ℏ G / c^{5})}^{1 / 2} = 5.4 \times 10^{- 44} \sec, \\ M^{*} & = (ℏ c / G)^{1 / 2} = 2.2 \times 10^{- 5} g, \end{aligned}

the only system of units, Planck (1899) pointed out, free, like black-body radiation itself, of all complications of solid-state physics, molecular binding, atomic constitution, and elementary particle structure, and drawing for its background only on the simplest and most universal principles of physics, the laws of gravitation and blackbody radiation. Relative to the Planck units, every constant in every other part of physics is expressed as a pure number.

Three hierarchies of fossils: molecules, nuclei, particles

Reason for identity in mass of particles of the same species?

Reprocessing of physical constants

No pure numbers in physics are more impressive than

ℏ c / e^{2} = 137.0360

and the so-called "big numbers" [Eddington (1931, 1936, 1946); Dirac (1937, 1938); Jordan (1955, 1959); Dicke (1959b, 1961, 1964b); Hayakawa (1965a,b); Carter (1968b)]:

\begin{aligned} \sim 10^{80} particles in the universe,* \\ \sim 10^{40} \sim \frac{10^{28} cm}{10^{- 12} cm} \sim \frac{{(\binom{radius of universe at}{maximum expansion})}^{*}}{(\begin{matrix} "size" of an elementary \\ particle \end{matrix}}, \\ \sim 10^{40} \sim \frac{e^{2}}{G m M} \sim \frac{(electric forces)}{(gravitational forces),} \\ \sim 10^{20} \sim \frac{e^{2} / m c^{2}}{{(ℏ G / c^{3})}^{1 / 2}} \sim \frac{(\begin{matrix} size" of an elementary \\ particle \end{matrix}}{(Planck length)}, \\ \sim 10^{10} \sim \frac{(\binom{number of photons}{in universe})}{(\begin{matrix} number of baryons \\ in universe \end{matrix}} . \end{aligned}

Some understanding of the relationships between these numbers has been won [Carter (1968b)]. Never has any explanation appeared for their enormous magnitude, nor will there ever, if the view is correct that reprocessing the universe reprocesses also the physical constants. These constants on that view are not part of the laws of physics. They are part of the initial-value data. Such numbers are freshly given for each fresh cycle of expansion of the universe. To look for a physical explanation for the "big numbers" would thus seem to be looking for the right answer to the wrong question.

In the week between one storm and the next, most features of the weather are ever-changing, but some special patterns of the wind last the week. If the term "frozen features of the meteorology" is appropriate for them, much more so would it seem appropriate for the big numbers, the physical constants and the spectrum of particle masses in the cycle between one reprocessing of the universe and another.

A per cent or so change one way in one of the "constants,"

ℏ c / e^{2}

, will cause all stars to be red stars; and a comparable change the other way will make all stars be blue stars, according to Carter (1968b). In neither case will any star like the sun be possible. He raises the question whether life could have developed if the determinants of the physical constants had differed substantially from those that characterize this cycle of the universe.

Dicke (1961) has pointed out that the right order of ideas may not be, here is the universe, so what must man be; but here is man, so what must the universe

be? In other words: (1) What good is a universe without awareness of that universe? But: (2) Awareness demands life. (3) Life demands the presence of elements heavier than hydrogen. (4) The production of heavy elements demands thermonuclear combustion. (5) Thermonuclear combustion normally requires several

10^{9}

years of cooking time in a star. (6) Several

10^{9}

years of time will not and cannot be available in a closed universe, according to general relativity, unless the radius-at-maximumexpansion of that universe is several

10^{9}

light years or more. So why on this view is the universe as big as it is? Because only so can man be here!

In brief, the considerations of Carter and Dicke would seem to raise the idea of the "biological selection of physical constants." However, to "select" is impossible unless there are options to select between. Exactly such options would seem for the first time to be held out by the only over-all picture of the gravitational collapse of the universe that one sees how to put forward today, the pregeometry black-box model of the reprocessing of the universe.

Proceeding with all caution into uncharted territory, one must nevertheless be aware that the conclusions one is reaching and the questions one is asking at a given stage of the analysis may be only stepping stones on the way to still more penetrating questions and an even more remarkable picture. To speak of "reprocessing and selection" may only be a halfway point on the road toward thinking of the universe as Leibniz did, as a world of relationships, not a world of machinery. Far from being brought into its present condition by "reprocessing" from earlier cycles, may the universe in some strange sense be "brought into being" by the participation of those who participate? On this view the concept of "cycles" would even seem to be altogether wrong. Instead the vital act is the act of participation. "Participator" is the incontrovertible new concept given by quantum mechanics; it strikes down the term "observer" of classical theory, the man who stands safely behind the thick glass wall and watches what goes on without taking part. It can't be done, quantum mechanics says. Even with the lowly electron one must participate before one can give any meaning whatsoever to its position or its momentum. Is this firmly established result the tiny tip of a giant iceberg? Does the universe also derive its meaning from "participation"? Are we destined to return to the great concept of Leibniz, of "preestablished harmony" ("Leibniz logic loop"), before we can make the next great advance?

Rich prospects stand open for investigation in gravitation physics, from neutron stars to cosmology and from post-Newtonian celestial mechanics to gravitational waves. Einstein's geometrodynamics exposes itself to destruction on a dozen fronts and by a thousand predictions. No predictions subject to early test are more entrancing than those on the formation and properties of a black hole, "laboratory model" for some of what is predicted for the universe itself. No field is more pregnant with the future than gravitational collapse. No more revolutionary views of man and the universe has one ever been driven to consider seriously than those that come out of pondering the paradox of collapse, the greatest crisis of physics of all time.

Black hole as "laboratory" model for collapse of universe

"Omnibus ex nihil ducendis sufficit unum!" (One suffices to create Everything of nothingl) Gottrried wilhelm von leibniz
(Sur l'air de Auprès de ma blonde)
Dans les jardins d'Asnières La science a refleuri
Tous les savants du monde
Apportent leurs écrits

Refrain:

Auprès de nos ondes Qu'il fait bon, fait bon, fait bon Auprès de nos ondes Qu'il fait bon rêver

Tous les savants du monde Apportent leurs écrits Loi gravitationnelle Sans tenseur d'énergie

Loi gravitationnelle
Sans tenseur d'énergie
De ravissants modèles
Pour la cosmologie
De ravissants modèles
Pour la cosmologie
Pour moi ne m'en faut guère
Car j'en ai un joli
Pour moi ne m'en faut guère Car j'en ai un joli
// est dans ma cervelle
Voici mon manuscrit
Le champ laisse des plumes Aux bosses de l'espace-temps En prendrons quelques unes Pour décrire le mouvement
C. CATTANEO, J. GÉHÉNIAU
M. MAVRIDES, and M. A. TONNELAT

Reprinied with the kind pemisssion of the guthors.

When Arthur Evans began this excavation neither he nor anyone knew that he would uncover an unknown world.

PAT (Mrs. Hypatia Vourloumis at Knossos (1971))

And as imagination bodies forth
The form of things unknown, the poet's pen
Turns them to shapes, and gives to airy nothing A local habitation and a name.
SHAKESPEARE

Appreciation and farewell to our patient reader.

Chaces ho Niene Kips. Jhome

*See Thorne and Will (1971), or Will (1972), for expositions of both frameworks and a comparison of them.
*R. H. Dicke calls this principle "The weak equivalence principle." We prefer to avoid confusion with the equivalence principle (Chapter 16).
$†$ In general relativity, one often uses an alternative definition of test body, which places no constraint on the self-gravitational energy [abandon condition (1) while retaining (2)]. Such a definition is preferable, in principle, because the theory of matter has not been developed sufficiently to decide whether (and no objective test has ever been proposed to decide whether), gravitational energy at the subnuclear scale is a small fraction, a large fraction, or the entirety of the rest mass. But for present purposes a definition constraining test bodies to have $M / R ≪ 1$ is preferable for two reasons. First, most theories of gravity that currently "compete" with Einstein's (a) agree with the principle of uniqueness of free fall when the macroscopic, Newtonian, self-gravitational energy is neglected $(M / R ≪ 1)$ , but (b) violate that principle when macroscopic, Newtonian self-gravitational energy is taken into account. See $$ 40.9$ for details. Second, the test bodies used in the Eobtvös-Dicke experiment have $M / R$ so small that their macroscopic, Newtonian, self-gravitational energies are, in fact, negligible ( $M / R \sim E_{grav} / M \sim 10^{- 27}$ ).
*For a 2 per cent test of time dilation with muons of ${(1 - v^{2})}^{- 1 / 2} \sim 12$ in a storage ring, see Farley, Bailey, Brown, Giesch, Jöstlein, van der Meer, Picasso, and Tannenbaum (1966). For earlier time-dilation experiments see Frisch and Smith (1963); Durbin, Loar, and Havens (1952); and Rossi and Hall (1941).
$†$ See p. 18 of Lichtenberg (1965) for a discussion of Lorentz invariance, spin and statistics, the TCP theorem, and relevant experiments.
- The experiment of Farley et al, is a 2 percent check of acceleration-independence of the muon decay rate for energies $E / m = {(1 - v^{2})}^{- 1 / 2} \sim 12$ and for accelerations, as measured in the muon rest frame, of $a = 5 \times 10^{20} cm / \sec^{2} = 0.6 {cm}^{- 1}$ .
*For a review of other, less-precise redshift experiments, see Bertotti, Brill, and Krotkov (1962).
*For a slightly narrower definition of metric theories, see Thorne and Will (1971).
$^{a}$ These heuristic descriptions are based on equations (39.23).
$^{b}$ For expositions of these theories see Box 39.1. For derivation of their PPN values and of PPN values for other theories, see Ni (1972).
*See, e.g., a long series of papers by Chandrasekhar and his associates in the Astrophysical Journal, beginning with Chandrasekhar (1965a,b,c).
*In the solar system, post-Newtonian corrections due to anisotropic stresses are so much smaller than other post-Newtonian corrections that there is no hope of measuring them in the 1970's. For this reason, elsewhere in the literature (but not in this book) the PPN formalism treats all stresses at the post-Newtonian level as isotropic pressures, thereby setting to zero the PPN parameter $η$ of $$ § 39.8 - 39.11$ .
*WARNING: Throughout the literature the notation $Φ$ is used where we use $Ψ$ for the functional $(39.23 g)$ , and $ϕ$ is used for our $ψ$ . We are forced to violate the standard notation to avoid confusion with the Newtonian potential $Φ = - U$ . However, we urge that nobody else violate the standard notation!
- Of course, from the point of view of Einstein's full general relativity theory, all that legalistically counts is the one and only curved-spacetime geometry of the real physical world. All these "individual fields" are mere bookkeepers' discourse, and they are best abandoned (they cease to be useful) when one passes from the post-Newtonian limit to the full Einstein theory.
$^{a}$ Here (observed delay)/(Einstein prediction) is the value of $\frac{1}{2} (1 + γ)$ obtained by fitting the observational data, $Δ τ (τ)$ , to a more sophisticated version of the PPN prediction (40.14). This more sophisticated version includes the gravitational influences of all the planets on the orbits of reflector and Earth: also the effect of the moon on the Earth's orbit and the effect of the Earth's rotation on the travel time; also, to as good an extent as possible, the delay due to dispersion in the solar corona and wind. "Formal standard error" and "one-sigma error" are defined in the table in Box 40.1.
*This analogy can be made mathematically rigorous; see footnote on p. 255 of Thorne (1971); see also, $$ 21.12$ on Mach's principle.
*The dragging of inertial frames by a rotating body plays important roles elsewhere in gravitation physics, e.g., in the definition of angular momentum for a gravitating body ( $$ 19.2$ ), and in black-hole physics (Chapter 33). The effect was first discussed and calculated by Thirring and Lense (1918). More recent calculations by Brill and Cohen (1966) of idealized situations where the effect may be large give insight into the mechanism of the effect. See also the discussion of Mach's principle in $$ 21.12$ .
*See any standard textbook for a description of Cavendish experiments. By his original version of the experiment, with two separated spheres suspended by fine wires, Henry Cavendish (1798) inferred the mass and hence the density of the Earth. He reported: "By a mean of the experiments made with the wire first used, the density of the Earth comes out 5.48 times greater than that of water; and by a mean of those made with the latter wire it comes out the same; and ... the extreme results do not differ from the mean more than 0.38 , or $1 / 14$ of the whole." The most precise method of measuring $G$ today [Rose et al. (1969)] gives $G_{C} = (6.674 \pm .004) \times 10^{- 8} {cm}^{3} / g \sec^{2}$ (one standard deviation).
*In the same city on June 21, 1972 President Eamon de Valera told one of the authors that, while in jail one evening in 1916, scheduled to be shot the next morning, he wrote down the formula of which he was so fond, $i^{2} = j^{2} = k^{2} = i j k = - 1$ .
*An "active" transformation changes one vector into another, while leaving unchanged the underlying reference frame (if there is one). By contrast, a "passive" transformation leaves all vectors unchanged, but alters the reference frame. All transformations in previous chapters of this book were passive.
$^{*}$ Actually $S \equiv S_{ADM} \equiv 16 π S_{true} = 16 π$ (true dynamic path length).
The Marchon lecture given by J. A. W. at the University of Newcastle-upon-Tyne, May 18, 1971, and the Nuffield lecture given at Cambridge University July 19, 1971, were based on the material presented in this chapter.
*Wheeler's story about the vizier and what the vizier had to say about superspace was told at the May 18, 1970, Gwatt Seminar on the Bearings of Topology upon General Relativity. Kilmister's (1971) published article alludes to the unpublished story, but does not actually contain it.
*Values based on the "typical cosmological model" of Box 27.4; subject to much uncertainty, in the present state of astrophysical distance determinations, not least because the latitude in these numbers is even enough to be compatible with an open universe.
All of these endeavors are based on the belief that existence should have a completely harmonious structure. Today we have less ground than ever before for allowing ourselves to be forced away from this wonderful belief.

снаттер 38

TESTING THE FOUNDATIONS OF RELATIVITY

§38.1. TESTING IS EASIER IN THE SOLAR SYSTEM THAN IN REMOTE SPACE

§38.2. THEORETICAL FRAMEWORKS FOR ANALYZING TESTS OF GENERAL RELATIVITY

The rest of this chapter is

§38.3. TESTS OF THE PRINCIPLE OF THE UNIQUENESS OF FREE FALL: EÖTVÖS-DICKE EXPERIMENT

Box 38.2 BARON LORAND VON EÖTVÖS

Box 38.3 ROBERT HENRY DICKE May 6, 1916, St. Louis, Missouri Cyrus Fogg Brackett Professor of Physics at Princeton University

§38.4. TESTS FOR THE EXISTENCE OF A METRIC GOVERNING LENGTH AND TIME MEASUREMENTS, AND PARTICLE KINEMATICS

§38.5. TESTS OF GEODESIC MOTION: GRAVITATIONAL REDSHIFT EXPERIMENTS

§38.6. TESTS OF THE EQUIVALENCE PRINCIPLE

§38.7. TESTS FOR THE EXISTENCE OF UNKNOWN LONG-RANGE FIELDS

OTHER THEORIES OF GRAVITY AND THE POST-NEWTONIAN APPROXIMATION

§39.1. OTHER THEORIES

§39.2. METRIC THEORIES OF GRAVITY

§39.3. POST-NEWTONIAN LIMIT AND PPN FORMALISM

Box 39.1 THE THEORIES OF DICKE-BRANS-JORDAN AND OF NI

A. Dicke-Brans-Jordan

B. Ni

Box 39.2 HEURISTIC DESCRIPTION OF THE TEN PPN PARAMETERS

Historical and Notational Notes

§39.4. PPN COORDINATE SYSTEM

§39.5. DESCRIPTION OF THE MATTER IN THE SOLAR SYSTEM

§39.6. NATURE OF THE POST-NEWTONIAN EXPANSION

EXERCISES

Exercise 39.1. ORDERS OF MAGNITUDE IN GRAVITATIONALLY BOUND SYSTEMS

Exercise 39.2. PATTERN OF TERMS IN POST-NEWTONIAN EXPANSION

§39.7. NEWTONIAN APPROXIMATION

Box 39.3 POST-NEWTONIAN EXPANSION OF THE METRIC COEFFICIENTS

EXERCISES

Exercise 39.3. NEWTONIAN APPROXIMATION

Exercise 39.4. A USEFUL FORMULA

Exercise 39.5. STRESS TENSOR FOR NEWTONIAN GRAVITATIONAL FIELD

Exercise 39.6. NEWTONIAN VIRIAL THEOREMS

Exercise 39.7. PULSATION FREQUENCY FOR NEWTONIAN STAR

§39.8. PPN METRIC COEFFICIENTS

EXERCISES

Exercise 39.8. ABSENCE OF "METRIC-GENERATES-METRIC" TERMS IN POST-NEWTONIAN LIMIT

Exercise 39.9. REMOVAL OF Σ Σ Sigma\boldsymbol{\Sigma}Σ TERM FROM g 00 g 00 g_(00)\boldsymbol{g}_{00}g00

§39.9. VELOCITY OF PPN COORDINATES RELATIVE TO "UNIVERSAL REST FRAME"

EXERCISE

Exercise 39.11. TRANSFORMATION TO MOVING FRAME

§39.10. PPN STRESS-ENERGY TENSOR

Exercise 39.12. THE TRANSFORMATION BETWEEN COMOVING FRAME AND PPN FRAME

§39.11. PPN EQUATIONS OF MOTION

Exercise 39.13. EQUATIONS OF MOTION

§39.12. RELATION OF PPN COORDINATES TO SURROUNDING UNIVERSE

§39.13. SUMMARY OF PPN FORMALISM

Exercise 39.15. MANY-BODY SYSTEM IN POST-NEWTONIAN LIMIT OF GENERAL RELATIVITY

Box 39.4 SUMMARY OF THE PPN FORMALISM

I. Variables

Box 39.5 PPN PARAMETERS USED IN LITERATURE: A TRANSLATOR'S GUIDE

B. Significance of Revised Parameters

сниетеп 40

SOLAR-SYSTEM EXPERIMENTS

§40.1. MANY EXPERIMENTS OPEN TO DISTINGUISH GENERAL RELATIVITY FROM PROPOSED METRIC THEORIES OF GRAVITY

Exercise 40.1. PPN METRIC FOR IDEALIZED SUN [Track 2]

EXERCISES

§40.2. THE USE OF LIGHT RAYS AND RADIO WAVES TO TEST GRAVITY

§40.3. '"LIGHT"' DEFLECTION

Exercise 40.2. TRAJECTORY OF LIGHT RAY IN SUN'S GRAVITATIONAL FIELD

EXERCISE

§40.4. TIME-DELAY IN RADAR PROPAGATION

Eclipse Measurements

Measurements on the Deflection of Radio Waves

EXERCISE

Exercise 40.3. FERMAT'S PRINCIPLE

Box 40.2 RADAR TIME DELAY IN THE SOLAR SYSTEM: EXPERIMENTAL RESULTS

§40.5. PERIHELION SHIFT AND PERIODIC PERTURBATIONS IN GEODESIC ORBITS

EXERCISES

Exercise 40.4. DERIVATION OF PERIHELION SHIFT IN PPN FORMALISM

Exercise 40.5. PERIHELION SHIFT FOR OBLATE SUN

§40.6. THREE-BODY EFFECTS IN THE LUNAR ORBIT

§40.7. THE DRAGGING OF INERTIAL FRAMES

EXERCISES

Exercise 40.6. PRECESSIONAL ANGULAR VELOCITY

Exercise 40.7. OFF-DIAGONAL TERMS IN METRIC ABOUT THE EARTH

Exercise 40.8. SPIN-CURVATURE COUPLING

§40.8. IS THE GRAVITATIONAL CONSTANT CONSTANT?

Exercise 40.9. CAVENDISH CONSTANT FOR IDEALIZED SUN

Exercise 39.9. REMOVAL OF $Σ$ TERM FROM $g_{00}$

§43.2. THE DYNAMICS OF GEOMETRY DESCRIBED IN THE LANGUAGE OF THE SUPERSPACE OF THE $^{(3)} y^{'}$ 's

$- 44$

Box 44.3 THE DIFFICULTIES WITH ATTEMPTS TO FIND A NATURAL PLACE FOR SPIN $\frac{1}{2}$ IN EINSTEIN'S STANDARD GEOMETRODYNAMICS